From Coq Require Import ssreflect ssrfun ssrbool.

From Coq.Logic Require Import FunctionalExtensionality PropExtensionality Classical_Pred_Type Classical_Prop.
From Coq.micromega Require Import Lia.
From Coq.Program Require Import Wf.

From Equations Require Import Equations.

From stdpp Require Import base fin_sets.
New coercion path [Bool.Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
From stdpp Require Import pmap gmap mapset fin_sets sets propset list_numbers.
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]

From MatchingLogic.Utils Require Import Lattice stdpp_ext extralibrary.
From MatchingLogic Require Import
  Syntax
  Freshness
  NamedAxioms
  IndexManipulation
.

Import MatchingLogic.Syntax.Notations.
Import MatchingLogic.Substitution.Notations.

(** ** Matching Logic Semantics *)
Section semantics.

  Context {signature : Signature}.
  Open Scope ml_scope.

  (* Model of AML ref. snapshot: Definition 2 *)

  Polymorphic Cumulative
  Record Model := {
    Domain :> Type;
    Domain_inhabited :> Inhabited Domain;
    app_interp : Domain -> Domain -> propset Domain;
    sym_interp (sigma : symbols) : propset Domain;
  }.

  Section with_model.
    Context {M : Model}.

  Definition Empty : propset (Domain M) := @empty (propset (Domain M)) _.
  Definition Full : propset (Domain M) := @top (propset (Domain M)) _.

  (* full set and empty set are distinct *)
  Lemma empty_impl_not_full : forall (S : propset (Domain M)),
      S = Empty -> S <> Full.
signature: Signature
M: Model
∀ S : propset M, S = Empty → S ≠ Full
  Proof.
signature: Signature
M: Model
∀ S : propset M, S = Empty → S ≠ Full
    intros S H.
signature: Signature
M: Model
S: propset M
H: S = Empty
S ≠ Full
    intros HContra.
signature: Signature
M: Model
S: propset M
H: S = Empty
HContra: S = Full
False rewrite -> HContra in H.
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
False
    assert (Hw1: (@inhabitant _ (Domain_inhabited M)) ∈ Full).
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
inhabitant ∈ Full
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
Hw1: inhabitant ∈ Full
False
    {
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
inhabitant ∈ Full unfold Full.
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
inhabitant ∈ ⊤ apply elem_of_top.
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
True exact I. }
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
Hw1: inhabitant ∈ Full
False
    rewrite H in Hw1.
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
Hw1: inhabitant ∈ Empty
False
    unfold Empty in Hw1.
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
Hw1: inhabitant ∈ ∅
False
    apply not_elem_of_empty in Hw1.
signature: Signature
M: Model
S: propset M
H: Full = Empty
HContra: S = Full
Hw1: False
False exact Hw1.
  Qed.

  Lemma full_impl_not_empty : forall (S : propset (Domain M)),
      S = Full -> S <> Empty.
signature: Signature
M: Model
∀ S : propset M, S = Full → S ≠ Empty
  Proof.
signature: Signature
M: Model
∀ S : propset M, S = Full → S ≠ Empty
    intros S H HContra.
signature: Signature
M: Model
S: propset M
H: S = Full
HContra: S = Empty
False
    rewrite -> HContra in H.
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
False
    assert (Hw1: (@inhabitant _ (Domain_inhabited M)) ∈ Full).
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
inhabitant ∈ Full
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
Hw1: inhabitant ∈ Full
False
    {
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
inhabitant ∈ Full unfold Full.
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
inhabitant ∈ ⊤ apply elem_of_top.
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
True exact I. }
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
Hw1: inhabitant ∈ Full
False
    rewrite <- H in Hw1.
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
Hw1: inhabitant ∈ Empty
False
    unfold Empty in Hw1.
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
Hw1: inhabitant ∈ ∅
False
    apply not_elem_of_empty in Hw1.
signature: Signature
M: Model
S: propset M
H: Empty = Full
HContra: S = Empty
Hw1: False
False exact Hw1.
  Qed.

  (* element and set variable valuations *)
  Polymorphic Record Valuation : Type := mkValuation
  { evar_valuation : evar -> Domain M ;
    svar_valuation : svar -> propset (Domain M) ;
  }.

  Definition update_evar_val
    (ev : evar) (x : Domain M) (val : Valuation) : Valuation :=
    {|
    evar_valuation := fun ev' : evar =>
      if decide (ev = ev') is left _ then x else evar_valuation val ev' ;
    svar_valuation := (svar_valuation val)
    |}.

  Definition update_svar_val
    (sv : svar) (X : propset (Domain M)) (val : Valuation) : Valuation :=
    {| 
    evar_valuation := (evar_valuation val);
    svar_valuation := fun sv' : svar =>
      if decide (sv = sv') is left _ then X else svar_valuation val sv' ;
    |}.

  Lemma update_evar_val_svar_val_comm
    (ev : evar) (x : Domain M)
    (sv : svar) (X : propset (Domain M))
    (ρ : Valuation) :
    update_evar_val ev x (update_svar_val sv X ρ)
    = update_svar_val sv X (update_evar_val ev x ρ).
signature: Signature
M: Model
ev: evar
x: M
sv: svar
X: propset M
ρ: Valuation
update_evar_val ev x (update_svar_val sv X ρ) =
update_svar_val sv X (update_evar_val ev x ρ)
  Proof.
signature: Signature
M: Model
ev: evar
x: M
sv: svar
X: propset M
ρ: Valuation
update_evar_val ev x (update_svar_val sv X ρ) =
update_svar_val sv X (update_evar_val ev x ρ)
    unfold update_evar_val, update_svar_val.
signature: Signature
M: Model
ev: evar
x: M
sv: svar
X: propset M
ρ: Valuation
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (ev = ev')
      then x
      else
       evar_valuation
         {|
           evar_valuation := evar_valuation ρ;
           svar_valuation :=
             λ sv' : svar,
               if decide (sv = sv')
               then X
               else svar_valuation ρ sv'
         |} ev';
  svar_valuation :=
    svar_valuation
      {|
        evar_valuation := evar_valuation ρ;
        svar_valuation :=
          λ sv' : svar,
            if decide (sv = sv')
            then X
            else svar_valuation ρ sv'
      |}
|} =
{|
  evar_valuation :=
    evar_valuation
      {|
        evar_valuation :=
          λ ev' : evar,
            if decide (ev = ev')
            then x
            else evar_valuation ρ ev';
        svar_valuation := svar_valuation ρ
      |};
  svar_valuation :=
    λ sv' : svar,
      if decide (sv = sv')
      then X
      else
       svar_valuation
         {|
           evar_valuation :=
             λ ev' : evar,
               if decide (ev = ev')
               then x
               else evar_valuation ρ ev';
           svar_valuation := svar_valuation ρ
         |} sv'
|} simpl.
signature: Signature
M: Model
ev: evar
x: M
sv: svar
X: propset M
ρ: Valuation
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (ev = ev')
      then x
      else evar_valuation ρ ev';
  svar_valuation :=
    λ sv' : svar,
      if decide (sv = sv')
      then X
      else svar_valuation ρ sv'
|} =
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (ev = ev')
      then x
      else evar_valuation ρ ev';
  svar_valuation :=
    λ sv' : svar,
      if decide (sv = sv')
      then X
      else svar_valuation ρ sv'
|}
    reflexivity.
  Qed.

  Lemma update_svar_val_comm :
    forall (X1 X2 : svar) (S1 S2 : propset (Domain M)) (ρ : Valuation),
      X1 <> X2 ->
      update_svar_val X1 S1 (update_svar_val X2 S2 ρ)
      = update_svar_val X2 S2 (update_svar_val X1 S1 ρ).
signature: Signature
M: Model
∀ (X1 X2 : svar) (S1 S2 : propset M) (ρ : Valuation),
  X1 ≠ X2
  → update_svar_val X1 S1 (update_svar_val X2 S2 ρ) =
    update_svar_val X2 S2 (update_svar_val X1 S1 ρ)
  Proof.
signature: Signature
M: Model
∀ (X1 X2 : svar) (S1 S2 : propset M) (ρ : Valuation),
  X1 ≠ X2
  → update_svar_val X1 S1 (update_svar_val X2 S2 ρ) =
    update_svar_val X2 S2 (update_svar_val X1 S1 ρ)
    intros.
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
update_svar_val X1 S1 (update_svar_val X2 S2 ρ) =
update_svar_val X2 S2 (update_svar_val X1 S1 ρ)
    unfold update_svar_val.
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
{|
  evar_valuation :=
    evar_valuation
      {|
        evar_valuation := evar_valuation ρ;
        svar_valuation :=
          λ sv' : svar,
            if decide (X2 = sv')
            then S2
            else svar_valuation ρ sv'
      |};
  svar_valuation :=
    λ sv' : svar,
      if decide (X1 = sv')
      then S1
      else
       svar_valuation
         {|
           evar_valuation := evar_valuation ρ;
           svar_valuation :=
             λ sv'0 : svar,
               if decide (X2 = sv'0)
               then S2
               else svar_valuation ρ sv'0
         |} sv'
|} =
{|
  evar_valuation :=
    evar_valuation
      {|
        evar_valuation := evar_valuation ρ;
        svar_valuation :=
          λ sv' : svar,
            if decide (X1 = sv')
            then S1
            else svar_valuation ρ sv'
      |};
  svar_valuation :=
    λ sv' : svar,
      if decide (X2 = sv')
      then S2
      else
       svar_valuation
         {|
           evar_valuation := evar_valuation ρ;
           svar_valuation :=
             λ sv'0 : svar,
               if decide (X1 = sv'0)
               then S1
               else svar_valuation ρ sv'0
         |} sv'
|} simpl.
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
{|
  evar_valuation := evar_valuation ρ;
  svar_valuation :=
    λ sv' : svar,
      if decide (X1 = sv')
      then S1
      else
       if decide (X2 = sv')
       then S2
       else svar_valuation ρ sv'
|} =
{|
  evar_valuation := evar_valuation ρ;
  svar_valuation :=
    λ sv' : svar,
      if decide (X2 = sv')
      then S2
      else
       if decide (X1 = sv')
       then S1
       else svar_valuation ρ sv'
|} f_equal.
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
(λ sv' : svar,
   if decide (X1 = sv')
   then S1
   else
    if decide (X2 = sv')
    then S2
    else svar_valuation ρ sv') =
(λ sv' : svar,
   if decide (X2 = sv')
   then S2
   else
    if decide (X1 = sv')
    then S1
    else svar_valuation ρ sv')
    apply functional_extensionality.
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
∀ x : svar,
  (if decide (X1 = x)
   then S1
   else
    if decide (X2 = x) then S2 else svar_valuation ρ x) =
  (if decide (X2 = x)
   then S2
   else
    if decide (X1 = x) then S1 else svar_valuation ρ x)
    intros.
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
x: svar
(if decide (X1 = x)
 then S1
 else
  if decide (X2 = x) then S2 else svar_valuation ρ x) =
(if decide (X2 = x)
 then S2
 else
  if decide (X1 = x) then S1 else svar_valuation ρ x)
    destruct (decide (X1 = x)),(decide (X2 = x)); subst.
signature: Signature
M: Model
S1, S2: propset M
ρ: Valuation
x: svar
H: x ≠ x
S1 = S2
signature: Signature
M: Model
X2: svar
S1, S2: propset M
ρ: Valuation
x: svar
H: x ≠ X2
n: X2 ≠ x
S1 = S1
signature: Signature
M: Model
X1: svar
S1, S2: propset M
ρ: Valuation
x: svar
H, n: X1 ≠ x
S2 = S2
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
x: svar
n: X1 ≠ x
n0: X2 ≠ x
svar_valuation ρ x = svar_valuation ρ x
    *
signature: Signature
M: Model
S1, S2: propset M
ρ: Valuation
x: svar
H: x ≠ x
S1 = S2 unfold not in H.
signature: Signature
M: Model
S1, S2: propset M
ρ: Valuation
x: svar
H: x = x → False
S1 = S2 assert (x = x).
signature: Signature
M: Model
S1, S2: propset M
ρ: Valuation
x: svar
H: x = x → False
x = x
signature: Signature
M: Model
S1, S2: propset M
ρ: Valuation
x: svar
H: x = x → False
H0: x = x
S1 = S2 reflexivity.
signature: Signature
M: Model
S1, S2: propset M
ρ: Valuation
x: svar
H: x = x → False
H0: x = x
S1 = S2 apply H in H0.
signature: Signature
M: Model
S1, S2: propset M
ρ: Valuation
x: svar
H: x = x → False
H0: False
S1 = S2 destruct H0.
    *
signature: Signature
M: Model
X2: svar
S1, S2: propset M
ρ: Valuation
x: svar
H: x ≠ X2
n: X2 ≠ x
S1 = S1 reflexivity.
    *
signature: Signature
M: Model
X1: svar
S1, S2: propset M
ρ: Valuation
x: svar
H, n: X1 ≠ x
S2 = S2 reflexivity.
    *
signature: Signature
M: Model
X1, X2: svar
S1, S2: propset M
ρ: Valuation
H: X1 ≠ X2
x: svar
n: X1 ≠ x
n0: X2 ≠ x
svar_valuation ρ x = svar_valuation ρ x reflexivity.
  Qed.

  Lemma update_svar_val_shadow : forall (X : svar)
                                        (S1 S2 : propset (Domain M))
                                        (ρ : Valuation),
      update_svar_val X S1 (update_svar_val X S2 ρ) = update_svar_val X S1 ρ.
signature: Signature
M: Model
∀ (X : svar) (S1 S2 : propset M) (ρ : Valuation),
  update_svar_val X S1 (update_svar_val X S2 ρ) =
  update_svar_val X S1 ρ
  Proof.
signature: Signature
M: Model
∀ (X : svar) (S1 S2 : propset M) (ρ : Valuation),
  update_svar_val X S1 (update_svar_val X S2 ρ) =
  update_svar_val X S1 ρ
    intros.
signature: Signature
M: Model
X: svar
S1, S2: propset M
ρ: Valuation
update_svar_val X S1 (update_svar_val X S2 ρ) =
update_svar_val X S1 ρ unfold update_svar_val.
signature: Signature
M: Model
X: svar
S1, S2: propset M
ρ: Valuation
{|
  evar_valuation :=
    evar_valuation
      {|
        evar_valuation := evar_valuation ρ;
        svar_valuation :=
          λ sv' : svar,
            if decide (X = sv')
            then S2
            else svar_valuation ρ sv'
      |};
  svar_valuation :=
    λ sv' : svar,
      if decide (X = sv')
      then S1
      else
       svar_valuation
         {|
           evar_valuation := evar_valuation ρ;
           svar_valuation :=
             λ sv'0 : svar,
               if decide (X = sv'0)
               then S2
               else svar_valuation ρ sv'0
         |} sv'
|} =
{|
  evar_valuation := evar_valuation ρ;
  svar_valuation :=
    λ sv' : svar,
      if decide (X = sv')
      then S1
      else svar_valuation ρ sv'
|} simpl.
signature: Signature
M: Model
X: svar
S1, S2: propset M
ρ: Valuation
{|
  evar_valuation := evar_valuation ρ;
  svar_valuation :=
    λ sv' : svar,
      if decide (X = sv')
      then S1
      else
       if decide (X = sv')
       then S2
       else svar_valuation ρ sv'
|} =
{|
  evar_valuation := evar_valuation ρ;
  svar_valuation :=
    λ sv' : svar,
      if decide (X = sv')
      then S1
      else svar_valuation ρ sv'
|}
    f_equal.
signature: Signature
M: Model
X: svar
S1, S2: propset M
ρ: Valuation
(λ sv' : svar,
   if decide (X = sv')
   then S1
   else
    if decide (X = sv')
    then S2
    else svar_valuation ρ sv') =
(λ sv' : svar,
   if decide (X = sv')
   then S1
   else svar_valuation ρ sv')
    apply functional_extensionality.
signature: Signature
M: Model
X: svar
S1, S2: propset M
ρ: Valuation
∀ x : svar,
  (if decide (X = x)
   then S1
   else
    if decide (X = x) then S2 else svar_valuation ρ x) =
  (if decide (X = x) then S1 else svar_valuation ρ x)
    intros.
signature: Signature
M: Model
X: svar
S1, S2: propset M
ρ: Valuation
x: svar
(if decide (X = x)
 then S1
 else
  if decide (X = x) then S2 else svar_valuation ρ x) =
(if decide (X = x) then S1 else svar_valuation ρ x) destruct (decide (X = x)); reflexivity.
  Qed.

  Lemma update_svar_val_neq (ρ : Valuation) X1 S X2 :
    X1 <> X2 -> svar_valuation (update_svar_val X1 S ρ) X2 = svar_valuation ρ X2.
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
X1 ≠ X2
→ svar_valuation (update_svar_val X1 S ρ) X2 =
  svar_valuation ρ X2
  Proof.
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
X1 ≠ X2
→ svar_valuation (update_svar_val X1 S ρ) X2 =
  svar_valuation ρ X2
    unfold update_svar_val.
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
X1 ≠ X2
→ svar_valuation
    {|
      evar_valuation := evar_valuation ρ;
      svar_valuation :=
        λ sv' : svar,
          if decide (X1 = sv')
          then S
          else svar_valuation ρ sv'
    |} X2 = svar_valuation ρ X2 intros.
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
H: X1 ≠ X2
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X1 = sv')
        then S
        else svar_valuation ρ sv'
  |} X2 = svar_valuation ρ X2 simpl.
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
H: X1 ≠ X2
(if decide (X1 = X2) then S else svar_valuation ρ X2) =
svar_valuation ρ X2
    destruct (decide (X1 = X2)); simpl.
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
H: X1 ≠ X2
e: X1 = X2
S = svar_valuation ρ X2
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
H, n: X1 ≠ X2
svar_valuation ρ X2 = svar_valuation ρ X2
    -
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
H: X1 ≠ X2
e: X1 = X2
S = svar_valuation ρ X2 contradiction.
    -
signature: Signature
M: Model
ρ: Valuation
X1: svar
S: propset M
X2: svar
H, n: X1 ≠ X2
svar_valuation ρ X2 = svar_valuation ρ X2 auto.
  Qed.
  
  Lemma update_evar_val_comm :
    forall (x1 x2 : evar) (m1 m2 : Domain M) (ρ : Valuation),
      x1 <> x2 ->
      update_evar_val x1 m1 (update_evar_val x2 m2 ρ)
      = update_evar_val x2 m2 (update_evar_val x1 m1 ρ).
signature: Signature
M: Model
∀ (x1 x2 : evar) (m1 m2 : M) (ρ : Valuation),
  x1 ≠ x2
  → update_evar_val x1 m1 (update_evar_val x2 m2 ρ) =
    update_evar_val x2 m2 (update_evar_val x1 m1 ρ)
  Proof.
signature: Signature
M: Model
∀ (x1 x2 : evar) (m1 m2 : M) (ρ : Valuation),
  x1 ≠ x2
  → update_evar_val x1 m1 (update_evar_val x2 m2 ρ) =
    update_evar_val x2 m2 (update_evar_val x1 m1 ρ)
    intros.
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
update_evar_val x1 m1 (update_evar_val x2 m2 ρ) =
update_evar_val x2 m2 (update_evar_val x1 m1 ρ)
    unfold update_evar_val.
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x1 = ev')
      then m1
      else
       evar_valuation
         {|
           evar_valuation :=
             λ ev'0 : evar,
               if decide (x2 = ev'0)
               then m2
               else evar_valuation ρ ev'0;
           svar_valuation := svar_valuation ρ
         |} ev';
  svar_valuation :=
    svar_valuation
      {|
        evar_valuation :=
          λ ev' : evar,
            if decide (x2 = ev')
            then m2
            else evar_valuation ρ ev';
        svar_valuation := svar_valuation ρ
      |}
|} =
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x2 = ev')
      then m2
      else
       evar_valuation
         {|
           evar_valuation :=
             λ ev'0 : evar,
               if decide (x1 = ev'0)
               then m1
               else evar_valuation ρ ev'0;
           svar_valuation := svar_valuation ρ
         |} ev';
  svar_valuation :=
    svar_valuation
      {|
        evar_valuation :=
          λ ev' : evar,
            if decide (x1 = ev')
            then m1
            else evar_valuation ρ ev';
        svar_valuation := svar_valuation ρ
      |}
|} simpl.
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x1 = ev')
      then m1
      else
       if decide (x2 = ev')
       then m2
       else evar_valuation ρ ev';
  svar_valuation := svar_valuation ρ
|} =
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x2 = ev')
      then m2
      else
       if decide (x1 = ev')
       then m1
       else evar_valuation ρ ev';
  svar_valuation := svar_valuation ρ
|} f_equal.
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
(λ ev' : evar,
   if decide (x1 = ev')
   then m1
   else
    if decide (x2 = ev')
    then m2
    else evar_valuation ρ ev') =
(λ ev' : evar,
   if decide (x2 = ev')
   then m2
   else
    if decide (x1 = ev')
    then m1
    else evar_valuation ρ ev')
    apply functional_extensionality.
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
∀ x : evar,
  (if decide (x1 = x)
   then m1
   else
    if decide (x2 = x) then m2 else evar_valuation ρ x) =
  (if decide (x2 = x)
   then m2
   else
    if decide (x1 = x) then m1 else evar_valuation ρ x)
    intros.
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
x: evar
(if decide (x1 = x)
 then m1
 else
  if decide (x2 = x) then m2 else evar_valuation ρ x) =
(if decide (x2 = x)
 then m2
 else
  if decide (x1 = x) then m1 else evar_valuation ρ x)
    destruct (decide (x1 = x)),(decide (x2 = x)); subst.
signature: Signature
M: Model
m1, m2: M
ρ: Valuation
x: evar
H: x ≠ x
m1 = m2
signature: Signature
M: Model
x2: evar
m1, m2: M
ρ: Valuation
x: evar
H: x ≠ x2
n: x2 ≠ x
m1 = m1
signature: Signature
M: Model
x1: evar
m1, m2: M
ρ: Valuation
x: evar
H, n: x1 ≠ x
m2 = m2
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
x: evar
n: x1 ≠ x
n0: x2 ≠ x
evar_valuation ρ x = evar_valuation ρ x
    *
signature: Signature
M: Model
m1, m2: M
ρ: Valuation
x: evar
H: x ≠ x
m1 = m2 unfold not in H.
signature: Signature
M: Model
m1, m2: M
ρ: Valuation
x: evar
H: x = x → False
m1 = m2 assert (x = x).
signature: Signature
M: Model
m1, m2: M
ρ: Valuation
x: evar
H: x = x → False
x = x
signature: Signature
M: Model
m1, m2: M
ρ: Valuation
x: evar
H: x = x → False
H0: x = x
m1 = m2 reflexivity.
signature: Signature
M: Model
m1, m2: M
ρ: Valuation
x: evar
H: x = x → False
H0: x = x
m1 = m2 apply H in H0.
signature: Signature
M: Model
m1, m2: M
ρ: Valuation
x: evar
H: x = x → False
H0: False
m1 = m2 destruct H0.
    *
signature: Signature
M: Model
x2: evar
m1, m2: M
ρ: Valuation
x: evar
H: x ≠ x2
n: x2 ≠ x
m1 = m1 reflexivity.
    *
signature: Signature
M: Model
x1: evar
m1, m2: M
ρ: Valuation
x: evar
H, n: x1 ≠ x
m2 = m2 reflexivity.
    *
signature: Signature
M: Model
x1, x2: evar
m1, m2: M
ρ: Valuation
H: x1 ≠ x2
x: evar
n: x1 ≠ x
n0: x2 ≠ x
evar_valuation ρ x = evar_valuation ρ x reflexivity.
  Qed.

  Lemma update_evar_val_shadow : forall (x : evar)
                                        (m1 m2 : Domain M)
                                        (ρ : Valuation),
      update_evar_val x m1 (update_evar_val x m2 ρ) = update_evar_val x m1 ρ.
signature: Signature
M: Model
∀ (x : evar) (m1 m2 : M) (ρ : Valuation),
  update_evar_val x m1 (update_evar_val x m2 ρ) =
  update_evar_val x m1 ρ
  Proof.
signature: Signature
M: Model
∀ (x : evar) (m1 m2 : M) (ρ : Valuation),
  update_evar_val x m1 (update_evar_val x m2 ρ) =
  update_evar_val x m1 ρ
    intros.
signature: Signature
M: Model
x: evar
m1, m2: M
ρ: Valuation
update_evar_val x m1 (update_evar_val x m2 ρ) =
update_evar_val x m1 ρ unfold update_evar_val.
signature: Signature
M: Model
x: evar
m1, m2: M
ρ: Valuation
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x = ev')
      then m1
      else
       evar_valuation
         {|
           evar_valuation :=
             λ ev'0 : evar,
               if decide (x = ev'0)
               then m2
               else evar_valuation ρ ev'0;
           svar_valuation := svar_valuation ρ
         |} ev';
  svar_valuation :=
    svar_valuation
      {|
        evar_valuation :=
          λ ev' : evar,
            if decide (x = ev')
            then m2
            else evar_valuation ρ ev';
        svar_valuation := svar_valuation ρ
      |}
|} =
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x = ev')
      then m1
      else evar_valuation ρ ev';
  svar_valuation := svar_valuation ρ
|} simpl.
signature: Signature
M: Model
x: evar
m1, m2: M
ρ: Valuation
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x = ev')
      then m1
      else
       if decide (x = ev')
       then m2
       else evar_valuation ρ ev';
  svar_valuation := svar_valuation ρ
|} =
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x = ev')
      then m1
      else evar_valuation ρ ev';
  svar_valuation := svar_valuation ρ
|} f_equal.
signature: Signature
M: Model
x: evar
m1, m2: M
ρ: Valuation
(λ ev' : evar,
   if decide (x = ev')
   then m1
   else
    if decide (x = ev')
    then m2
    else evar_valuation ρ ev') =
(λ ev' : evar,
   if decide (x = ev')
   then m1
   else evar_valuation ρ ev') apply functional_extensionality.
signature: Signature
M: Model
x: evar
m1, m2: M
ρ: Valuation
∀ x0 : evar,
  (if decide (x = x0)
   then m1
   else
    if decide (x = x0)
    then m2
    else evar_valuation ρ x0) =
  (if decide (x = x0) then m1 else evar_valuation ρ x0)
    intros.
signature: Signature
M: Model
x: evar
m1, m2: M
ρ: Valuation
x0: evar
(if decide (x = x0)
 then m1
 else
  if decide (x = x0) then m2 else evar_valuation ρ x0) =
(if decide (x = x0) then m1 else evar_valuation ρ x0) destruct (decide (x = x0)); reflexivity.
  Qed.

  Lemma update_evar_val_same x m ρ :
    evar_valuation (update_evar_val x m ρ) x = m.
signature: Signature
M: Model
x: evar
m: M
ρ: Valuation
evar_valuation (update_evar_val x m ρ) x = m
  Proof.
signature: Signature
M: Model
x: evar
m: M
ρ: Valuation
evar_valuation (update_evar_val x m ρ) x = m
    unfold update_evar_val.
signature: Signature
M: Model
x: evar
m: M
ρ: Valuation
evar_valuation
  {|
    evar_valuation :=
      λ ev' : evar,
        if decide (x = ev')
        then m
        else evar_valuation ρ ev';
    svar_valuation := svar_valuation ρ
  |} x = m simpl.
signature: Signature
M: Model
x: evar
m: M
ρ: Valuation
(if decide (x = x) then m else evar_valuation ρ x) = m
    rewrite decide_eq_same.
signature: Signature
M: Model
x: evar
m: M
ρ: Valuation
m = m reflexivity.
  Qed.

  Lemma update_evar_val_neq (ρ : Valuation) x1 e x2 :
    x1 <> x2 -> evar_valuation (update_evar_val x1 e ρ) x2 = evar_valuation ρ x2.
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
x1 ≠ x2
→ evar_valuation (update_evar_val x1 e ρ) x2 =
  evar_valuation ρ x2
  Proof.
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
x1 ≠ x2
→ evar_valuation (update_evar_val x1 e ρ) x2 =
  evar_valuation ρ x2
    unfold update_evar_val.
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
x1 ≠ x2
→ evar_valuation
    {|
      evar_valuation :=
        λ ev' : evar,
          if decide (x1 = ev')
          then e
          else evar_valuation ρ ev';
      svar_valuation := svar_valuation ρ
    |} x2 = evar_valuation ρ x2 simpl.
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
x1 ≠ x2
→ (if decide (x1 = x2) then e else evar_valuation ρ x2) =
  evar_valuation ρ x2 intros.
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
H: x1 ≠ x2
(if decide (x1 = x2) then e else evar_valuation ρ x2) =
evar_valuation ρ x2
    destruct (decide (x1 = x2)); simpl.
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
H: x1 ≠ x2
e0: x1 = x2
e = evar_valuation ρ x2
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
H, n: x1 ≠ x2
evar_valuation ρ x2 = evar_valuation ρ x2
    -
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
H: x1 ≠ x2
e0: x1 = x2
e = evar_valuation ρ x2 contradiction.
    -
signature: Signature
M: Model
ρ: Valuation
x1: evar
e: M
x2: evar
H, n: x1 ≠ x2
evar_valuation ρ x2 = evar_valuation ρ x2 auto.
  Qed.

  Lemma update_evar_val_same_2 x ρ :
    update_evar_val x (evar_valuation ρ x) ρ = ρ.
signature: Signature
M: Model
x: evar
ρ: Valuation
update_evar_val x (evar_valuation ρ x) ρ = ρ
  Proof.
signature: Signature
M: Model
x: evar
ρ: Valuation
update_evar_val x (evar_valuation ρ x) ρ = ρ
    destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
update_evar_val x
  (evar_valuation
     {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
     x)
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} =
{| evar_valuation := ρₑ; svar_valuation := ρₛ |} simpl.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
update_evar_val x (ρₑ x)
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} =
{| evar_valuation := ρₑ; svar_valuation := ρₛ |} unfold update_evar_val.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x = ev')
      then ρₑ x
      else
       evar_valuation
         {|
           evar_valuation := ρₑ; svar_valuation := ρₛ
         |} ev';
  svar_valuation :=
    svar_valuation
      {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
|} = {| evar_valuation := ρₑ; svar_valuation := ρₛ |} simpl.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
{|
  evar_valuation :=
    λ ev' : evar,
      if decide (x = ev') then ρₑ x else ρₑ ev';
  svar_valuation := ρₛ
|} = {| evar_valuation := ρₑ; svar_valuation := ρₛ |} f_equal.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
(λ ev' : evar,
   if decide (x = ev') then ρₑ x else ρₑ ev') = ρₑ
    apply functional_extensionality.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
∀ x0 : evar,
  (if decide (x = x0) then ρₑ x else ρₑ x0) = ρₑ x0 intros x0.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
x0: evar
(if decide (x = x0) then ρₑ x else ρₑ x0) = ρₑ x0
    unfold update_evar_val.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
x0: evar
(if decide (x = x0) then ρₑ x else ρₑ x0) = ρₑ x0 destruct (decide (x = x0)); simpl.
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
x0: evar
e: x = x0
ρₑ x = ρₑ x0
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
x0: evar
n: x ≠ x0
ρₑ x0 = ρₑ x0
    -
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
x0: evar
e: x = x0
ρₑ x = ρₑ x0 subst.
signature: Signature
M: Model
ρₑ: evar → M
ρₛ: svar → propset M
x0: evar
ρₑ x0 = ρₑ x0 reflexivity.
    -
signature: Signature
M: Model
x: evar
ρₑ: evar → M
ρₛ: svar → propset M
x0: evar
n: x ≠ x0
ρₑ x0 = ρₑ x0 reflexivity.
  Qed.

  Lemma update_svar_val_same X S ρ :
    svar_valuation (update_svar_val X S ρ) X = S.
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
svar_valuation (update_svar_val X S ρ) X = S
  Proof.
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
svar_valuation (update_svar_val X S ρ) X = S
    unfold update_svar_val.
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then S
        else svar_valuation ρ sv'
  |} X = S simpl.
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
(if decide (X = X) then S else svar_valuation ρ X) = S destruct (decide (X = X)); simpl.
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
e: X = X
S = S
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
n: X ≠ X
svar_valuation ρ X = S
    +
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
e: X = X
S = S reflexivity.
    +
signature: Signature
M: Model
X: svar
S: propset M
ρ: Valuation
n: X ≠ X
svar_valuation ρ X = S contradiction.
  Qed.

  Lemma update_svar_val_same_2 x ρ :
    update_svar_val x (svar_valuation ρ x) ρ = ρ.
signature: Signature
M: Model
x: svar
ρ: Valuation
update_svar_val x (svar_valuation ρ x) ρ = ρ
  Proof.
signature: Signature
M: Model
x: svar
ρ: Valuation
update_svar_val x (svar_valuation ρ x) ρ = ρ
    destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
update_svar_val x
  (svar_valuation
     {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
     x)
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} =
{| evar_valuation := ρₑ; svar_valuation := ρₛ |} simpl.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
update_svar_val x (ρₛ x)
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} =
{| evar_valuation := ρₑ; svar_valuation := ρₛ |} unfold update_svar_val.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
{|
  evar_valuation :=
    evar_valuation
      {| evar_valuation := ρₑ; svar_valuation := ρₛ |};
  svar_valuation :=
    λ sv' : svar,
      if decide (x = sv')
      then ρₛ x
      else
       svar_valuation
         {|
           evar_valuation := ρₑ; svar_valuation := ρₛ
         |} sv'
|} = {| evar_valuation := ρₑ; svar_valuation := ρₛ |} simpl.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
{|
  evar_valuation := ρₑ;
  svar_valuation :=
    λ sv' : svar,
      if decide (x = sv') then ρₛ x else ρₛ sv'
|} = {| evar_valuation := ρₑ; svar_valuation := ρₛ |} f_equal.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
(λ sv' : svar,
   if decide (x = sv') then ρₛ x else ρₛ sv') = ρₛ
    apply functional_extensionality.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
∀ x0 : svar,
  (if decide (x = x0) then ρₛ x else ρₛ x0) = ρₛ x0 intros x0.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
x0: svar
(if decide (x = x0) then ρₛ x else ρₛ x0) = ρₛ x0
    unfold update_svar_val.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
x0: svar
(if decide (x = x0) then ρₛ x else ρₛ x0) = ρₛ x0 destruct (decide (x = x0)); simpl.
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
x0: svar
e: x = x0
ρₛ x = ρₛ x0
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
x0: svar
n: x ≠ x0
ρₛ x0 = ρₛ x0
    -
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
x0: svar
e: x = x0
ρₛ x = ρₛ x0 subst.
signature: Signature
M: Model
ρₑ: evar → M
ρₛ: svar → propset M
x0: svar
ρₛ x0 = ρₛ x0 reflexivity.
    -
signature: Signature
M: Model
x: svar
ρₑ: evar → M
ρₛ: svar → propset M
x0: svar
n: x ≠ x0
ρₛ x0 = ρₛ x0 reflexivity.
  Qed.


  (* We use propositional extensionality here. *)
  #[export]
  Instance propset_leibniz_equiv : LeibnizEquiv (propset (Domain M)).
signature: Signature
M: Model
LeibnizEquiv (propset M)
  Proof.
signature: Signature
M: Model
LeibnizEquiv (propset M)
    intros x y H.
signature: Signature
M: Model
x, y: propset M
H: x ≡ y
x = y unfold equiv in H.
signature: Signature
M: Model
x, y: propset M
H: set_equiv_instance x y
x = y unfold set_equiv_instance in H.
signature: Signature
M: Model
x, y: propset M
H: ∀ x0 : M, x0 ∈ x ↔ x0 ∈ y
x = y
    destruct x,y.
signature: Signature
M: Model
propset_car, propset_car0: M → Prop
H: ∀ x : M,
  x ∈ {[ x | propset_car x ]}
  ↔ x ∈ {[ x | propset_car0 x ]}
{[ x | propset_car x ]} = {[ x | propset_car0 x ]}
    apply f_equal.
signature: Signature
M: Model
propset_car, propset_car0: M → Prop
H: ∀ x : M,
  x ∈ {[ x | propset_car x ]}
  ↔ x ∈ {[ x | propset_car0 x ]}
propset_car = propset_car0 apply functional_extensionality.
signature: Signature
M: Model
propset_car, propset_car0: M → Prop
H: ∀ x : M,
  x ∈ {[ x | propset_car x ]}
  ↔ x ∈ {[ x | propset_car0 x ]}
∀ x : M, propset_car x = propset_car0 x
    intros x.
signature: Signature
M: Model
propset_car, propset_car0: M → Prop
H: ∀ x : M,
  x ∈ {[ x | propset_car x ]}
  ↔ x ∈ {[ x | propset_car0 x ]}
x: M
propset_car x = propset_car0 x apply propositional_extensionality.
signature: Signature
M: Model
propset_car, propset_car0: M → Prop
H: ∀ x : M,
  x ∈ {[ x | propset_car x ]}
  ↔ x ∈ {[ x | propset_car0 x ]}
x: M
propset_car x ↔ propset_car0 x
    specialize (H x).
signature: Signature
M: Model
propset_car, propset_car0: M → Prop
x: M
H: x ∈ {[ x | propset_car x ]}
↔ x ∈ {[ x | propset_car0 x ]}
propset_car x ↔ propset_car0 x destruct H as [H1 H2].
signature: Signature
M: Model
propset_car, propset_car0: M → Prop
x: M
H1: x ∈ {[ x | propset_car x ]}
→ x ∈ {[ x | propset_car0 x ]}
H2: x ∈ {[ x | propset_car0 x ]}
→ x ∈ {[ x | propset_car x ]}
propset_car x ↔ propset_car0 x
    split; auto.
  Qed.

  (* extending pointwise application *)
  Polymorphic
  Definition app_ext
             (l r : propset (Domain M)) :
    propset (Domain M) :=
    PropSet (fun (e : (Domain M)) => exists (le re : (Domain M)), le ∈ l /\ re ∈ r /\ e ∈ (app_interp _) le re).

  Lemma app_ext_bot_r :
      forall S : propset (Domain M),
        app_ext S ∅ = ∅.
signature: Signature
M: Model
∀ S : propset M, app_ext S ∅ = ∅
  Proof.
signature: Signature
M: Model
∀ S : propset M, app_ext S ∅ = ∅
    intros S.
signature: Signature
M: Model
S: propset M
app_ext S ∅ = ∅ unfold app_ext.
signature: Signature
M: Model
S: propset M
{[ e | ∃ le re : M,
         le ∈ S ∧ re ∈ ∅ ∧ e ∈ app_interp M le re ]} =
∅
    rewrite -> elem_of_equiv_empty_L.
signature: Signature
M: Model
S: propset M
∀ x : M,
  x
  ∉ {[ e | ∃ le re : M,
             le ∈ S ∧ re ∈ ∅ ∧ e ∈ app_interp M le re ]}
    intros x Hcontra.
signature: Signature
M: Model
S: propset M
x: M
Hcontra: x
∈ {[ e | ∃ le re : M,
           le ∈ S
           ∧ re ∈ ∅ ∧ e ∈ app_interp M le re ]}
False rewrite -> elem_of_PropSet in Hcontra.
signature: Signature
M: Model
S: propset M
x: M
Hcontra: ∃ le re : M,
  le ∈ S ∧ re ∈ ∅ ∧ x ∈ app_interp M le re
False
    destruct Hcontra as [le [re [H1 [H2 H3] ] ] ].
signature: Signature
M: Model
S: propset M
x, le, re: M
H1: le ∈ S
H2: re ∈ ∅
H3: x ∈ app_interp M le re
False
    apply not_elem_of_empty in H2.
signature: Signature
M: Model
S: propset M
x, le, re: M
H1: le ∈ S
H2: False
H3: x ∈ app_interp M le re
False exact H2.
  Qed.

  Lemma app_ext_bot_l :
      forall S : propset (Domain M),
        app_ext ∅ S = ∅.
signature: Signature
M: Model
∀ S : propset M, app_ext ∅ S = ∅
  Proof.
signature: Signature
M: Model
∀ S : propset M, app_ext ∅ S = ∅
    intros S.
signature: Signature
M: Model
S: propset M
app_ext ∅ S = ∅ unfold app_ext.
signature: Signature
M: Model
S: propset M
{[ e | ∃ le re : M,
         le ∈ ∅ ∧ re ∈ S ∧ e ∈ app_interp M le re ]} =
∅
    rewrite -> elem_of_equiv_empty_L.
signature: Signature
M: Model
S: propset M
∀ x : M,
  x
  ∉ {[ e | ∃ le re : M,
             le ∈ ∅ ∧ re ∈ S ∧ e ∈ app_interp M le re ]}
    intros x Hcontra.
signature: Signature
M: Model
S: propset M
x: M
Hcontra: x
∈ {[ e | ∃ le re : M,
           le ∈ ∅
           ∧ re ∈ S ∧ e ∈ app_interp M le re ]}
False rewrite -> elem_of_PropSet in Hcontra.
signature: Signature
M: Model
S: propset M
x: M
Hcontra: ∃ le re : M,
  le ∈ ∅ ∧ re ∈ S ∧ x ∈ app_interp M le re
False
    destruct Hcontra as [le [re [H1 [H2 H3] ] ] ].
signature: Signature
M: Model
S: propset M
x, le, re: M
H1: le ∈ ∅
H2: re ∈ S
H3: x ∈ app_interp M le re
False
    apply not_elem_of_empty in H1.
signature: Signature
M: Model
S: propset M
x, le, re: M
H1: False
H2: re ∈ S
H3: x ∈ app_interp M le re
False exact H1.
  Qed.

  Lemma app_ext_monotonic_l :
      forall (S1 S2 S : propset (Domain M)),
        S1 ⊆ S2 -> (app_ext S1 S) ⊆ (app_ext S2 S).
signature: Signature
M: Model
∀ S1 S2 S : propset M,
  S1 ⊆ S2 → app_ext S1 S ⊆ app_ext S2 S
  Proof.
signature: Signature
M: Model
∀ S1 S2 S : propset M,
  S1 ⊆ S2 → app_ext S1 S ⊆ app_ext S2 S
    intros S1 S2 S H.
signature: Signature
M: Model
S1, S2, S: propset M
H: S1 ⊆ S2
app_ext S1 S ⊆ app_ext S2 S rewrite -> elem_of_subseteq in H.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
app_ext S1 S ⊆ app_ext S2 S
    rewrite -> elem_of_subseteq.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
∀ x : M, x ∈ app_ext S1 S → x ∈ app_ext S2 S intros x H'.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
x: M
H': x ∈ app_ext S1 S
x ∈ app_ext S2 S
    unfold app_ext in H'.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
x: M
H': x
∈ {[ e | ∃ le re : M,
           le ∈ S1
           ∧ re ∈ S ∧ e ∈ app_interp M le re ]}
x ∈ app_ext S2 S
    rewrite -> elem_of_PropSet in H'.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
x: M
H': ∃ le re : M,
  le ∈ S1 ∧ re ∈ S ∧ x ∈ app_interp M le re
x ∈ app_ext S2 S
    destruct H' as [le [re [H1 [H2 H3] ] ] ].
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
x, le, re: M
H1: le ∈ S1
H2: re ∈ S
H3: x ∈ app_interp M le re
x ∈ app_ext S2 S
    unfold app_ext.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
x, le, re: M
H1: le ∈ S1
H2: re ∈ S
H3: x ∈ app_interp M le re
x
∈ {[ e | ∃ le re : M,
           le ∈ S2 ∧ re ∈ S ∧ e ∈ app_interp M le re ]}
    rewrite -> elem_of_PropSet.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
x, le, re: M
H1: le ∈ S1
H2: re ∈ S
H3: x ∈ app_interp M le re
∃ le re : M, le ∈ S2 ∧ re ∈ S ∧ x ∈ app_interp M le re
    exists le,re.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S1 → x ∈ S2
x, le, re: M
H1: le ∈ S1
H2: re ∈ S
H3: x ∈ app_interp M le re
le ∈ S2 ∧ re ∈ S ∧ x ∈ app_interp M le re firstorder.
  Qed.

  Lemma app_ext_monotonic_r :
      forall (S S1 S2 : propset (Domain M)),
        S1 ⊆ S2 -> (app_ext S S1) ⊆ (app_ext S S2).
signature: Signature
M: Model
∀ S S1 S2 : propset M,
  S1 ⊆ S2 → app_ext S S1 ⊆ app_ext S S2
  Proof.
signature: Signature
M: Model
∀ S S1 S2 : propset M,
  S1 ⊆ S2 → app_ext S S1 ⊆ app_ext S S2
    intros S1 S2 S H.
signature: Signature
M: Model
S1, S2, S: propset M
H: S2 ⊆ S
app_ext S1 S2 ⊆ app_ext S1 S rewrite -> elem_of_subseteq in H.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
app_ext S1 S2 ⊆ app_ext S1 S
    rewrite -> elem_of_subseteq.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
∀ x : M, x ∈ app_ext S1 S2 → x ∈ app_ext S1 S intros x H'.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
x: M
H': x ∈ app_ext S1 S2
x ∈ app_ext S1 S
    unfold app_ext in H'.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
x: M
H': x
∈ {[ e | ∃ le re : M,
           le ∈ S1
           ∧ re ∈ S2 ∧ e ∈ app_interp M le re ]}
x ∈ app_ext S1 S
    rewrite -> elem_of_PropSet in H'.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
x: M
H': ∃ le re : M,
  le ∈ S1 ∧ re ∈ S2 ∧ x ∈ app_interp M le re
x ∈ app_ext S1 S
    destruct H' as [le [re [H1 [H2 H3] ] ] ].
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
x, le, re: M
H1: le ∈ S1
H2: re ∈ S2
H3: x ∈ app_interp M le re
x ∈ app_ext S1 S
    unfold app_ext.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
x, le, re: M
H1: le ∈ S1
H2: re ∈ S2
H3: x ∈ app_interp M le re
x
∈ {[ e | ∃ le re : M,
           le ∈ S1 ∧ re ∈ S ∧ e ∈ app_interp M le re ]}
    rewrite -> elem_of_PropSet.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
x, le, re: M
H1: le ∈ S1
H2: re ∈ S2
H3: x ∈ app_interp M le re
∃ le re : M, le ∈ S1 ∧ re ∈ S ∧ x ∈ app_interp M le re
    exists le,re.
signature: Signature
M: Model
S1, S2, S: propset M
H: ∀ x : M, x ∈ S2 → x ∈ S
x, le, re: M
H1: le ∈ S1
H2: re ∈ S2
H3: x ∈ app_interp M le re
le ∈ S1 ∧ re ∈ S ∧ x ∈ app_interp M le re firstorder.
  Qed.


  (* Semantics of AML ref. snapshot: Definition 3 *)

    Let OS := PropsetOrderedSet (Domain M).
    Let  L := PowersetLattice (Domain M).

    Equations? eval (ρ : Valuation) (ϕ : Pattern)
      : propset (@Domain M) by wf (size ϕ) :=
    eval ρ (patt_free_evar x)  := {[ evar_valuation ρ x ]} ;
    eval ρ (patt_free_svar X)  := svar_valuation ρ X ;
    eval ρ (patt_bound_evar n) := ∅ ;
    eval ρ (patt_bound_svar n) := ∅ ;
    eval ρ (patt_sym s)        := @sym_interp M s ;
    eval ρ (patt_app ϕ₁ ϕ₂)    := app_ext (eval ρ ϕ₁) (eval ρ ϕ₂) ;
    eval ρ (patt_bott)         := ∅ ;
    eval ρ (patt_imp ϕ₁ ϕ₂)    := (difference ⊤ (eval ρ ϕ₁)) ∪ (eval ρ ϕ₂) ;
    eval ρ (patt_exists ϕ')    := let x := fresh_evar ϕ' in
                                  propset_fa_union
                                  (fun e =>
                                    let ρ' := (update_evar_val x e ρ) in
                                    eval ρ' (ϕ'^{evar: 0 ↦ x})
                                  ) ;
    eval ρ (patt_mu ϕ')        := let X := fresh_svar ϕ' in
                                  @LeastFixpointOf _ OS L
                                  (fun S =>
                                    let ρ' := (update_svar_val X S ρ) in
                                    eval ρ' (ϕ'^{svar: 0 ↦ X})
                                  ) .
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ $ ϕ₂) → propset M
size ϕ₁ < size (ϕ₁ $ ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ $ ϕ₂) → propset M
size ϕ₂ < size (ϕ₁ $ ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ ---> ϕ₂) → propset M
size ϕ₁ < size (ϕ₁ ---> ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ ---> ϕ₂) → propset M
size ϕ₂ < size (ϕ₁ ---> ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ex , ϕ') → propset M
x:= fresh_evar ϕ': evar
e: M
ρ':= update_evar_val x e ρ: Valuation
size ϕ'^{evar:0↦x} < size (ex , ϕ')
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (mu , ϕ') → propset M
X:= fresh_svar ϕ': svar
S: propset M
ρ':= update_svar_val X S ρ: Valuation
size ϕ'^{svar:0↦X} < size (mu , ϕ')
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ $ ϕ₂) → propset M
size ϕ₁ < size (ϕ₁ $ ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ $ ϕ₂) → propset M
size ϕ₂ < size (ϕ₁ $ ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ ---> ϕ₂) → propset M
size ϕ₁ < size (ϕ₁ ---> ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ϕ₁ ---> ϕ₂) → propset M
size ϕ₂ < size (ϕ₁ ---> ϕ₂)
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ex , ϕ') → propset M
x:= fresh_evar ϕ': evar
e: M
ρ':= update_evar_val x e ρ: Valuation
size ϕ'^{evar:0↦x} < size (ex , ϕ')
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (mu , ϕ') → propset M
X:= fresh_svar ϕ': svar
S: propset M
ρ':= update_svar_val X S ρ: Valuation
size ϕ'^{svar:0↦X} < size (mu , ϕ')
      all: simpl; try lia.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ex , ϕ') → propset M
x:= fresh_evar ϕ': evar
e: M
ρ':= update_evar_val x e ρ: Valuation
size ϕ'^{evar:0↦x} < S (size ϕ')
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (mu , ϕ') → propset M
X:= fresh_svar ϕ': svar
S: propset M
ρ':= update_svar_val X S ρ: Valuation
size ϕ'^{svar:0↦X} < Datatypes.S (size ϕ')
      {
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ex , ϕ') → propset M
x:= fresh_evar ϕ': evar
e: M
ρ':= update_evar_val x e ρ: Valuation
size ϕ'^{evar:0↦x} < S (size ϕ') rewrite -evar_open_size.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (ex , ϕ') → propset M
x:= fresh_evar ϕ': evar
e: M
ρ':= update_evar_val x e ρ: Valuation
size ϕ' < S (size ϕ') lia. }
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (mu , ϕ') → propset M
X:= fresh_svar ϕ': svar
S: propset M
ρ':= update_svar_val X S ρ: Valuation
size ϕ'^{svar:0↦X} < Datatypes.S (size ϕ')
      {
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (mu , ϕ') → propset M
X:= fresh_svar ϕ': svar
S: propset M
ρ':= update_svar_val X S ρ: Valuation
size ϕ'^{svar:0↦X} < Datatypes.S (size ϕ') rewrite -svar_open_size.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval: Valuation
→ ∀ x0 : Pattern,
    size x0 < size (mu , ϕ') → propset M
X:= fresh_svar ϕ': svar
S: propset M
ρ':= update_svar_val X S ρ: Valuation
size ϕ' < Datatypes.S (size ϕ') lia. }
    Defined.

    Definition Fassoc ρ ϕ X :=
      λ S, eval (update_svar_val X S ρ) ϕ.

    Lemma eval_free_evar_simpl
          (ρ : Valuation)
          (x : evar) :
      eval ρ (patt_free_evar x) = {[ evar_valuation ρ x ]}.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
x: evar
eval ρ (patt_free_evar x) = {[evar_valuation ρ x]}
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
x: evar
eval ρ (patt_free_evar x) = {[evar_valuation ρ x]} simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
x: evar
{[evar_valuation ρ x]} = {[evar_valuation ρ x]} reflexivity. Qed.

    Lemma eval_free_svar_simpl
          (ρ : Valuation)
          (X : svar) :
      eval ρ (patt_free_svar X) = svar_valuation ρ X.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
X: svar
eval ρ (patt_free_svar X) = svar_valuation ρ X
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
X: svar
eval ρ (patt_free_svar X) = svar_valuation ρ X simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
X: svar
svar_valuation ρ X = svar_valuation ρ X reflexivity. Qed.

    Lemma eval_bound_evar_simpl
          (ρ : Valuation)
          (x : db_index) :
      eval ρ (patt_bound_evar x) = ∅.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
x: db_index
eval ρ (patt_bound_evar x) = ∅
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
x: db_index
eval ρ (patt_bound_evar x) = ∅ simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
x: db_index
∅ = ∅ reflexivity. Qed.

    Lemma eval_bound_svar_simpl
          (ρ : Valuation)
          (X : db_index) :
      eval ρ (patt_bound_svar X) = ∅.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
X: db_index
eval ρ (patt_bound_svar X) = ∅
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
X: db_index
eval ρ (patt_bound_svar X) = ∅ simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
X: db_index
∅ = ∅ reflexivity. Qed.

    Lemma eval_sym_simpl
          (ρ : Valuation)
          (s : symbols) :
      eval ρ (patt_sym s) = @sym_interp M s.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
s: symbols
eval ρ (patt_sym s) = sym_interp M s
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
s: symbols
eval ρ (patt_sym s) = sym_interp M s simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
s: symbols
sym_interp M s = sym_interp M s reflexivity. Qed.

    Lemma eval_app_simpl
          (ρ : Valuation)
          (ϕ₁ ϕ₂ : Pattern) :
      eval ρ (patt_app ϕ₁ ϕ₂) = app_ext (eval ρ ϕ₁) (eval ρ ϕ₂).
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval ρ (ϕ₁ $ ϕ₂) = app_ext (eval ρ ϕ₁) (eval ρ ϕ₂)
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval ρ (ϕ₁ $ ϕ₂) = app_ext (eval ρ ϕ₁) (eval ρ ϕ₂) simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
app_ext (eval ρ ϕ₁) (eval ρ ϕ₂) =
app_ext (eval ρ ϕ₁) (eval ρ ϕ₂) reflexivity. Qed.

    Lemma eval_bott_simpl
          (ρ : Valuation) :
      eval ρ patt_bott = ∅.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
eval ρ ⊥ = ∅
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
eval ρ ⊥ = ∅ simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
∅ = ∅ reflexivity. Qed.

    Lemma eval_imp_simpl
          (ρ : Valuation)
          (ϕ₁ ϕ₂ : Pattern) :
      eval ρ (patt_imp ϕ₁ ϕ₂) = (difference ⊤ (eval ρ ϕ₁)) ∪ (eval ρ ϕ₂).
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval ρ (ϕ₁ ---> ϕ₂) = ⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
eval ρ (ϕ₁ ---> ϕ₂) = ⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ₁, ϕ₂: Pattern
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ reflexivity. Qed.

    Lemma eval_ex_simpl
          (ρ : Valuation)
          (ϕ' : Pattern) :
      eval ρ (patt_exists ϕ') =
      let x := fresh_evar ϕ' in
      propset_fa_union
      (fun e =>
        let ρ' := (update_evar_val x e ρ) in
        eval ρ' (ϕ'^{evar: 0 ↦ x})
      ).
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval ρ (ex , ϕ') =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ' ϕ'^{evar:0↦x}))
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval ρ (ex , ϕ') =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ' ϕ'^{evar:0↦x})) simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ' ϕ'^{evar:0↦x})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ' ϕ'^{evar:0↦x})) reflexivity. Qed.

    Lemma eval_mu_simpl
          (ρ : Valuation)
          (ϕ' : Pattern) :
      eval ρ (patt_mu ϕ') =
      let X := fresh_svar ϕ' in
      @LeastFixpointOf _ OS L
      (fun S =>
        let ρ' := (update_svar_val X S ρ) in
        eval ρ' (ϕ'^{svar: 0 ↦ X})
      ).
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval ρ (mu , ϕ') =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' ϕ'^{svar:0↦X}))
    Proof.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
eval ρ (mu , ϕ') =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' ϕ'^{svar:0↦X})) simp eval.
signature: Signature
M: Model
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
ρ: Valuation
ϕ': Pattern
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' ϕ'^{svar:0↦X})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' ϕ'^{svar:0↦X})) reflexivity. Qed.

    (* TODO extend with derived constructs using typeclasses *)
    Definition eval_simpl :=
      ( eval_free_evar_simpl,
        eval_free_svar_simpl,
        eval_bound_evar_simpl,
        eval_bound_svar_simpl,
        eval_sym_simpl,  
        eval_app_simpl,
        eval_bott_simpl,
        eval_imp_simpl,
        eval_ex_simpl,
        eval_mu_simpl
      ).

End with_model.

Section with_explicit_model.
  Context (M : Model).
  (* Model predicate. Useful mainly if the pattern is well-formed. *)
  Definition M_predicate (ϕ : Pattern) : Prop := forall ρ,
      @eval M ρ ϕ = ⊤ \/ eval ρ ϕ = ∅.

  Lemma M_predicate_impl ϕ₁ ϕ₂ : M_predicate ϕ₁ -> M_predicate ϕ₂ -> M_predicate (patt_imp ϕ₁ ϕ₂).
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
M_predicate ϕ₁
→ M_predicate ϕ₂ → M_predicate (ϕ₁ ---> ϕ₂)
  Proof.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
M_predicate ϕ₁
→ M_predicate ϕ₂ → M_predicate (ϕ₁ ---> ϕ₂)
    unfold M_predicate.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
(∀ ρ : Valuation, eval ρ ϕ₁ = ⊤ ∨ eval ρ ϕ₁ = ∅)
→ (∀ ρ : Valuation, eval ρ ϕ₂ = ⊤ ∨ eval ρ ϕ₂ = ∅)
  → ∀ ρ : Valuation,
      eval ρ (ϕ₁ ---> ϕ₂) = ⊤
      ∨ eval ρ (ϕ₁ ---> ϕ₂) = ∅ intros Hp1 Hp2 ρ.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
Hp1: ∀ ρ : Valuation, eval ρ ϕ₁ = ⊤ ∨ eval ρ ϕ₁ = ∅
Hp2: ∀ ρ : Valuation, eval ρ ϕ₂ = ⊤ ∨ eval ρ ϕ₂ = ∅
ρ: Valuation
eval ρ (ϕ₁ ---> ϕ₂) = ⊤ ∨ eval ρ (ϕ₁ ---> ϕ₂) = ∅
    specialize (Hp1 ρ).
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hp1: eval ρ ϕ₁ = ⊤ ∨ eval ρ ϕ₁ = ∅
Hp2: ∀ ρ : Valuation, eval ρ ϕ₂ = ⊤ ∨ eval ρ ϕ₂ = ∅
eval ρ (ϕ₁ ---> ϕ₂) = ⊤ ∨ eval ρ (ϕ₁ ---> ϕ₂) = ∅ specialize (Hp2 ρ).
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hp1: eval ρ ϕ₁ = ⊤ ∨ eval ρ ϕ₁ = ∅
Hp2: eval ρ ϕ₂ = ⊤ ∨ eval ρ ϕ₂ = ∅
eval ρ (ϕ₁ ---> ϕ₂) = ⊤ ∨ eval ρ (ϕ₁ ---> ϕ₂) = ∅
    rewrite -> eval_imp_simpl.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hp1: eval ρ ϕ₁ = ⊤ ∨ eval ρ ϕ₁ = ∅
Hp2: eval ρ ϕ₂ = ⊤ ∨ eval ρ ϕ₂ = ∅
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
∨ ⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ∅
    destruct Hp1, Hp2; rewrite -> H; rewrite -> H0.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ⊤
H0: eval ρ ϕ₂ = ⊤
⊤ ∖ ⊤ ∪ ⊤ = ⊤ ∨ ⊤ ∖ ⊤ ∪ ⊤ = ∅
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ⊤
H0: eval ρ ϕ₂ = ∅
⊤ ∖ ⊤ ∪ ∅ = ⊤ ∨ ⊤ ∖ ⊤ ∪ ∅ = ∅
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ∅
H0: eval ρ ϕ₂ = ⊤
⊤ ∖ ∅ ∪ ⊤ = ⊤ ∨ ⊤ ∖ ∅ ∪ ⊤ = ∅
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ∅
H0: eval ρ ϕ₂ = ∅
⊤ ∖ ∅ ∪ ∅ = ⊤ ∨ ⊤ ∖ ∅ ∪ ∅ = ∅
    +
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ⊤
H0: eval ρ ϕ₂ = ⊤
⊤ ∖ ⊤ ∪ ⊤ = ⊤ ∨ ⊤ ∖ ⊤ ∪ ⊤ = ∅ left.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ⊤
H0: eval ρ ϕ₂ = ⊤
⊤ ∖ ⊤ ∪ ⊤ = ⊤ set_solver by fail.
    +
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ⊤
H0: eval ρ ϕ₂ = ∅
⊤ ∖ ⊤ ∪ ∅ = ⊤ ∨ ⊤ ∖ ⊤ ∪ ∅ = ∅ right.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ⊤
H0: eval ρ ϕ₂ = ∅
⊤ ∖ ⊤ ∪ ∅ = ∅ set_solver by fail.
    +
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ∅
H0: eval ρ ϕ₂ = ⊤
⊤ ∖ ∅ ∪ ⊤ = ⊤ ∨ ⊤ ∖ ∅ ∪ ⊤ = ∅ left.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ∅
H0: eval ρ ϕ₂ = ⊤
⊤ ∖ ∅ ∪ ⊤ = ⊤ set_solver by fail.
    +
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ∅
H0: eval ρ ϕ₂ = ∅
⊤ ∖ ∅ ∪ ∅ = ⊤ ∨ ⊤ ∖ ∅ ∪ ∅ = ∅ left.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H: eval ρ ϕ₁ = ∅
H0: eval ρ ϕ₂ = ∅
⊤ ∖ ∅ ∪ ∅ = ⊤ set_solver by fail.
  Qed.

  Hint Resolve M_predicate_impl : core.

  Lemma M_predicate_bott : M_predicate patt_bott.
signature: Signature
M: Model
M_predicate ⊥
  Proof.
signature: Signature
M: Model
M_predicate ⊥
    unfold M_predicate.
signature: Signature
M: Model
∀ ρ : Valuation, eval ρ ⊥ = ⊤ ∨ eval ρ ⊥ = ∅ intros.
signature: Signature
M: Model
ρ: Valuation
eval ρ ⊥ = ⊤ ∨ eval ρ ⊥ = ∅ right.
signature: Signature
M: Model
ρ: Valuation
eval ρ ⊥ = ∅
    apply eval_bott_simpl.
  Qed.

  Hint Resolve M_predicate_bott : core.

  Lemma M_predicate_exists ϕ :
    let x := evar_fresh (elements (free_evars ϕ)) in
    M_predicate (ϕ^{evar: 0 ↦ x}) -> M_predicate (patt_exists ϕ).
signature: Signature
M: Model
ϕ: Pattern
let x := evar_fresh (elements (free_evars ϕ)) in
M_predicate ϕ^{evar:0↦x} → M_predicate (ex , ϕ)
  Proof.
signature: Signature
M: Model
ϕ: Pattern
let x := evar_fresh (elements (free_evars ϕ)) in
M_predicate ϕ^{evar:0↦x} → M_predicate (ex , ϕ)
    simpl.
signature: Signature
M: Model
ϕ: Pattern
M_predicate
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))}
→ M_predicate (ex , ϕ) unfold M_predicate.
signature: Signature
M: Model
ϕ: Pattern
(∀ ρ : Valuation,
   eval ρ
     ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
   ⊤
   ∨ eval ρ
       ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
     ∅)
→ ∀ ρ : Valuation,
    eval ρ (ex , ϕ) = ⊤ ∨ eval ρ (ex , ϕ) = ∅ intros.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
eval ρ (ex , ϕ) = ⊤ ∨ eval ρ (ex , ϕ) = ∅
    rewrite -> eval_ex_simpl.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
(let x := fresh_evar ϕ in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ' ϕ^{evar:0↦x})) = ⊤
∨ (let x := fresh_evar ϕ in
   propset_fa_union
     (λ e : M,
        let ρ' := update_evar_val x e ρ in
        eval ρ' ϕ^{evar:0↦x})) = ∅
    simpl.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ⊤
∨ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) = ∅
    pose proof (H' := classic
                        (exists e : Domain M,
                            (eval
                               (update_evar_val (evar_fresh (elements (free_evars ϕ))) e ρ)
                               (ϕ^{evar: 0 ↦ evar_fresh (elements (free_evars ϕ))})
                            = ⊤))).
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H': (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
∨ ¬ (∃ e : M,
       eval
         (update_evar_val
            (evar_fresh (elements (free_evars ϕ)))
            e ρ)
         ϕ^{evar:0↦evar_fresh
                     (elements (free_evars ϕ))} =
       ⊤)
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ⊤
∨ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) = ∅
    destruct H'.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ⊤
∨ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) = ∅
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ⊤
∨ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) = ∅
    - (* For some member, the subformula evaluates to full set. *)
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ⊤
∨ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) = ∅
      left.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ⊤
      apply set_eq_subseteq.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ⊤
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      split.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ⊤
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
⊤
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      {
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ⊤ set_solver by fail. }
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
⊤
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      apply elem_of_subseteq.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
∀ x : M,
  x ∈ ⊤
  → x
    ∈ propset_fa_union
        (λ e : M,
           eval (update_evar_val (fresh_evar ϕ) e ρ)
             ϕ^{evar:0↦fresh_evar ϕ})
      intros x _.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh
                (elements (free_evars ϕ))} = ⊤
x: M
x
∈ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      destruct H0 as [x0 H0].
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
x0: M
H0: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) x0 ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
x: M
x
∈ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) unfold fresh_evar.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
x0: M
H0: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) x0 ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
x: M
x
∈ propset_fa_union
    (λ e : M,
       eval
         (update_evar_val
            (evar_fresh_s (free_evars ϕ)) e ρ)
         ϕ^{evar:0↦evar_fresh_s (free_evars ϕ)})
      unfold propset_fa_union.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
x0: M
H0: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) x0 ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
x: M
x
∈ {[ x | ∃ c : M,
           x
           ∈ eval
               (update_evar_val
                  (evar_fresh_s (free_evars ϕ)) c ρ)
               ϕ^{evar:0↦evar_fresh_s (free_evars ϕ)} ]} rewrite -> elem_of_PropSet.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
x0: M
H0: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) x0 ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
x: M
∃ c : M,
  x
  ∈ eval
      (update_evar_val (evar_fresh_s (free_evars ϕ)) c
         ρ) ϕ^{evar:0↦evar_fresh_s (free_evars ϕ)}
      exists x0.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
x0: M
H0: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) x0 ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
x: M
x
∈ eval
    (update_evar_val (evar_fresh_s (free_evars ϕ)) x0
       ρ) ϕ^{evar:0↦evar_fresh_s (free_evars ϕ)} rewrite H0.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
x0: M
H0: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) x0 ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
x: M
x ∈ ⊤ apply elem_of_top'.
    - (* The subformula does not evaluate to full set for any member. *)
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ⊤
∨ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) = ∅
      right.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ∅
      apply set_eq_subseteq.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      split.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      2: {
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) apply empty_subseteq. }
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
      apply elem_of_subseteq.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
∀ x : M,
  x
  ∈ propset_fa_union
      (λ e : M,
         eval (update_evar_val (fresh_evar ϕ) e ρ)
           ϕ^{evar:0↦fresh_evar ϕ}) → x ∈ ∅
      intros x H1.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
x: M
H1: x
∈ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
x ∈ ∅
      unfold propset_fa_union in H1.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
x: M
H1: x
∈ {[ x | ∃ c : M,
           x
           ∈ eval
               (update_evar_val 
                  (fresh_evar ϕ) c ρ)
               ϕ^{evar:0↦
               fresh_evar ϕ} ]}
x ∈ ∅
      rewrite -> elem_of_PropSet in H1.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
x: M
H1: ∃ c : M,
  x
  ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
      ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅
      destruct H1 as [c H1].
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: ¬ (∃ e : M,
     eval
       (update_evar_val
          (evar_fresh (elements (free_evars ϕ)))
          e ρ)
       ϕ^{evar:0↦evar_fresh
                   (elements (free_evars ϕ))} = ⊤)
x, c: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅
      unfold not in H0.
signature: Signature
M: Model
ϕ: Pattern
H: ∀ ρ : Valuation,
  eval ρ
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤
  ∨ eval ρ
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))} = ∅
ρ: Valuation
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x, c: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅
      specialize (H (update_evar_val (evar_fresh (elements (free_evars ϕ))) c ρ)).
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
∨ eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) c ρ)
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ∅
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅
      destruct H.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
∅
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅
      +
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅ exfalso.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
False apply H0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
∃ e : M,
  eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) e ρ)
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
  ⊤ exists c.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
⊤
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} = ⊤ apply H.
      +
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))} =
∅
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅ apply set_eq_subseteq in H.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))}
⊆ ∅
⊆ eval
    (update_evar_val
       (evar_fresh (elements (free_evars ϕ))) c ρ)
    ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))}
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅ destruct H as [H _].
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: eval
  (update_evar_val
     (evar_fresh (elements (free_evars ϕ))) c ρ)
  ϕ^{evar:0↦evar_fresh (elements (free_evars ϕ))}
⊆ ∅
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅
        rewrite ->  elem_of_subseteq in H.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
c: M
H: ∀ x : M,
  x
  ∈ eval
      (update_evar_val
         (evar_fresh (elements (free_evars ϕ))) c
         ρ)
      ϕ^{evar:0↦evar_fresh
                  (elements (free_evars ϕ))}
  → x ∈ ∅
H0: (∃ e : M,
   eval
     (update_evar_val
        (evar_fresh (elements (free_evars ϕ))) e
        ρ)
     ϕ^{evar:0↦evar_fresh
                 (elements (free_evars ϕ))} = ⊤)
→ False
x: M
H1: x
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x ∈ ∅ auto.
  Qed.

  Hint Resolve M_predicate_exists : core.

  Lemma predicate_not_empty_iff_full ϕ ρ :
    M_predicate ϕ ->
    @eval M ρ ϕ <> ∅ <->
    eval ρ ϕ = ⊤.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
M_predicate ϕ → eval ρ ϕ ≠ ∅ ↔ eval ρ ϕ = ⊤
  Proof.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
M_predicate ϕ → eval ρ ϕ ≠ ∅ ↔ eval ρ ϕ = ⊤
    intros Hmp.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ ≠ ∅ ↔ eval ρ ϕ = ⊤
    split.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ ≠ ∅ → eval ρ ϕ = ⊤
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ = ⊤ → eval ρ ϕ ≠ ∅
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ ≠ ∅ → eval ρ ϕ = ⊤ intros Hne.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
Hne: eval ρ ϕ ≠ ∅
eval ρ ϕ = ⊤ specialize (Hmp ρ).
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: eval ρ ϕ = ⊤ ∨ eval ρ ϕ = ∅
Hne: eval ρ ϕ ≠ ∅
eval ρ ϕ = ⊤
      destruct Hmp.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ⊤
Hne: eval ρ ϕ ≠ ∅
eval ρ ϕ = ⊤
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ∅
Hne: eval ρ ϕ ≠ ∅
eval ρ ϕ = ⊤
      +
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ⊤
Hne: eval ρ ϕ ≠ ∅
eval ρ ϕ = ⊤ assumption.
      +
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ∅
Hne: eval ρ ϕ ≠ ∅
eval ρ ϕ = ⊤ contradiction.
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ = ⊤ → eval ρ ϕ ≠ ∅ intros Hf.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
Hf: eval ρ ϕ = ⊤
eval ρ ϕ ≠ ∅
      apply full_impl_not_empty.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
Hf: eval ρ ϕ = ⊤
eval ρ ϕ = Full
      assumption.
  Qed.

  Lemma predicate_not_full_iff_empty ϕ ρ :
    M_predicate ϕ ->
    @eval M ρ ϕ <> ⊤ <->
    eval ρ ϕ = ∅.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
M_predicate ϕ → eval ρ ϕ ≠ ⊤ ↔ eval ρ ϕ = ∅
  Proof.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
M_predicate ϕ → eval ρ ϕ ≠ ⊤ ↔ eval ρ ϕ = ∅
    intros Hmp.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ ≠ ⊤ ↔ eval ρ ϕ = ∅
    split.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ ≠ ⊤ → eval ρ ϕ = ∅
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ = ∅ → eval ρ ϕ ≠ ⊤
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ ≠ ⊤ → eval ρ ϕ = ∅ intros Hne.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
Hne: eval ρ ϕ ≠ ⊤
eval ρ ϕ = ∅ specialize (Hmp ρ).
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: eval ρ ϕ = ⊤ ∨ eval ρ ϕ = ∅
Hne: eval ρ ϕ ≠ ⊤
eval ρ ϕ = ∅
      destruct Hmp.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ⊤
Hne: eval ρ ϕ ≠ ⊤
eval ρ ϕ = ∅
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ∅
Hne: eval ρ ϕ ≠ ⊤
eval ρ ϕ = ∅
      +
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ⊤
Hne: eval ρ ϕ ≠ ⊤
eval ρ ϕ = ∅ contradiction.
      +
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
H: eval ρ ϕ = ∅
Hne: eval ρ ϕ ≠ ⊤
eval ρ ϕ = ∅ assumption.
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
eval ρ ϕ = ∅ → eval ρ ϕ ≠ ⊤ intros Hf.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
Hf: eval ρ ϕ = ∅
eval ρ ϕ ≠ ⊤
      apply empty_impl_not_full.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
Hmp: M_predicate ϕ
Hf: eval ρ ϕ = ∅
eval ρ ϕ = Empty
      assumption.
  Qed.

  Lemma eval_impl_MP ϕ₁ ϕ₂ ρ :
    @eval M ρ (patt_imp ϕ₁ ϕ₂) = ⊤ ->
    eval ρ ϕ₁ = ⊤ ->
    eval ρ ϕ₂ = ⊤.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
eval ρ (ϕ₁ ---> ϕ₂) = ⊤
→ eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
  Proof.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
eval ρ (ϕ₁ ---> ϕ₂) = ⊤
→ eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
    unfold Full.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
eval ρ (ϕ₁ ---> ϕ₂) = ⊤
→ eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
    rewrite eval_imp_simpl.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
→ eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
    intros H1 H2.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H1: ⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
H2: eval ρ ϕ₁ = ⊤
eval ρ ϕ₂ = ⊤
    rewrite -> H2 in H1.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
H1: ⊤ ∖ ⊤ ∪ eval ρ ϕ₂ = ⊤
H2: eval ρ ϕ₁ = ⊤
eval ρ ϕ₂ = ⊤
    set_solver by auto.
  Qed.

  Lemma eval_predicate_impl ϕ₁ ϕ₂ ρ :
    M_predicate ϕ₁ ->
    @eval M ρ (patt_imp ϕ₁ ϕ₂) = ⊤
    <-> (eval ρ ϕ₁ = ⊤
         -> eval ρ ϕ₂ = ⊤).
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
M_predicate ϕ₁
→ eval ρ (ϕ₁ ---> ϕ₂) = ⊤
  ↔ (eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤)
  Proof.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
M_predicate ϕ₁
→ eval ρ (ϕ₁ ---> ϕ₂) = ⊤
  ↔ (eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤)
    intros Hpred.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
eval ρ (ϕ₁ ---> ϕ₂) = ⊤
↔ (eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤)
    split.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
eval ρ (ϕ₁ ---> ϕ₂) = ⊤
→ eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
(eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤)
→ eval ρ (ϕ₁ ---> ϕ₂) = ⊤
    -
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
eval ρ (ϕ₁ ---> ϕ₂) = ⊤
→ eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤ intros H H1.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ (ϕ₁ ---> ϕ₂) = ⊤
H1: eval ρ ϕ₁ = ⊤
eval ρ ϕ₂ = ⊤
      apply (eval_impl_MP _ _ _ H H1).
    -
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
(eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤)
→ eval ρ (ϕ₁ ---> ϕ₂) = ⊤ intros H.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
eval ρ (ϕ₁ ---> ϕ₂) = ⊤
      rewrite -> eval_imp_simpl.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
      destruct (classic (eval ρ ϕ₁ = ⊤)).
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ⊤
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ ≠ ⊤
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
      +
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ⊤
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤ specialize (H H0).
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ⊤
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
        rewrite -> H.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ⊤
⊤ ∖ eval ρ ϕ₁ ∪ ⊤ = ⊤ rewrite -> H0.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ⊤
⊤ ∖ ⊤ ∪ ⊤ = ⊤
        set_solver by fail.
      +
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ ≠ ⊤
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤ apply predicate_not_full_iff_empty in H0.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ∅
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ ≠ ⊤
H1: ∀ (ϕ : Pattern) (ρ : Valuation),
  M_predicate ϕ → eval ρ ϕ ≠ ⊤ → eval ρ ϕ = ∅
M_predicate ϕ₁ 2: apply Hpred.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ∅
⊤ ∖ eval ρ ϕ₁ ∪ eval ρ ϕ₂ = ⊤
        rewrite -> H0.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
ρ: Valuation
Hpred: M_predicate ϕ₁
H: eval ρ ϕ₁ = ⊤ → eval ρ ϕ₂ = ⊤
H0: eval ρ ϕ₁ = ∅
⊤ ∖ ∅ ∪ eval ρ ϕ₂ = ⊤
        set_solver by fail.
  Qed.
  
  (* ϕ is a well-formed body of ex *)
  Lemma eval_exists_predicate_full ϕ ρ :
    let x := fresh_evar ϕ in
    M_predicate (ϕ^{evar: 0 ↦ x}) ->
    eval ρ (patt_exists ϕ) = ⊤ <->
    ∃ (m : Domain M), eval (update_evar_val x m ρ) (ϕ^{evar: 0 ↦ x}) = ⊤.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
let x := fresh_evar ϕ in
M_predicate ϕ^{evar:0↦x}
→ eval ρ (ex , ϕ) = ⊤
  ↔ (∃ m : M,
       eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
  Proof.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
let x := fresh_evar ϕ in
M_predicate ϕ^{evar:0↦x}
→ eval ρ (ex , ϕ) = ⊤
  ↔ (∃ m : M,
       eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
    intros x Hpred.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
eval ρ (ex , ϕ) = ⊤
↔ (∃ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
    pose proof (Hpredex := M_predicate_exists _ Hpred).
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
eval ρ (ex , ϕ) = ⊤
↔ (∃ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
    rewrite -[eval _ _ = ⊤]predicate_not_empty_iff_full.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
M_predicate (ex , ϕ)
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
eval ρ (ex , ϕ) ≠ ∅
↔ (∃ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
    {
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
M_predicate (ex , ϕ) apply Hpredex. }
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
eval ρ (ex , ϕ) ≠ ∅
↔ (∃ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
    
    (*
        (* TODO: I would like to simplify the RHS, but cannot find a way. *)
        under [fun m => _]functional_extensionality => m.
        (* This fails *)
        Fail rewrite <- predicate_not_empty_iff_full.
        Fail rewrite -[_ = Full]predicate_not_empty_iff_full.
        over.
     *)


    rewrite -> Not_Empty_iff_Contains_Elements.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
(∃ x : M, x ∈ eval ρ (ex , ϕ))
↔ (∃ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
    split.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
(∃ x : M, x ∈ eval ρ (ex , ϕ))
→ ∃ m : M,
    eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
(∃ m : M,
   eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
→ ∃ x : M, x ∈ eval ρ (ex , ϕ)
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
(∃ x : M, x ∈ eval ρ (ex , ϕ))
→ ∃ m : M,
    eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤ intros [x0 Hx0].
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0: M
Hx0: x0 ∈ eval ρ (ex , ϕ)
∃ m : M, eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤
      rewrite -> eval_ex_simpl in Hx0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0: M
Hx0: x0
∈ (let x := fresh_evar ϕ in
   propset_fa_union
     (λ e : M,
        let ρ' := update_evar_val x e ρ in
        eval ρ' ϕ^{evar:0↦x}))
∃ m : M, eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤
      simpl in Hx0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0: M
Hx0: x0
∈ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
∃ m : M, eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤ unfold propset_fa_union in Hx0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0: M
Hx0: x0
∈ {[ x | ∃ c : M,
           x
           ∈ eval
               (update_evar_val 
                  (fresh_evar ϕ) c ρ)
               ϕ^{evar:0↦
               fresh_evar ϕ} ]}
∃ m : M, eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤ rewrite -> elem_of_PropSet in Hx0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0: M
Hx0: ∃ c : M,
  x0
  ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
      ϕ^{evar:0↦fresh_evar ϕ}
∃ m : M, eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤
      destruct Hx0 as [c Hc].
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
∃ m : M, eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤
      exists c.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
eval (update_evar_val x c ρ) ϕ^{evar:0↦x} = ⊤
      rewrite -predicate_not_empty_iff_full.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
M_predicate ϕ^{evar:0↦x}
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
eval (update_evar_val x c ρ) ϕ^{evar:0↦x} ≠ ∅
      {
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
M_predicate ϕ^{evar:0↦x} apply Hpred. }
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
eval (update_evar_val x c ρ) ϕ^{evar:0↦x} ≠ ∅
      rewrite Not_Empty_iff_Contains_Elements.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
∃ x0 : M,
  x0 ∈ eval (update_evar_val x c ρ) ϕ^{evar:0↦x}
      exists x0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
x0, c: M
Hc: x0
∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x0 ∈ eval (update_evar_val x c ρ) ϕ^{evar:0↦x} apply Hc.
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
(∃ m : M,
   eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤)
→ ∃ x : M, x ∈ eval ρ (ex , ϕ) intros [m Hm].
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m: M
Hm: eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ⊤
∃ x : M, x ∈ eval ρ (ex , ϕ)
      apply full_impl_not_empty in Hm.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m: M
Hm: eval (update_evar_val x m ρ) ϕ^{evar:0↦x} ≠ Empty
∃ x : M, x ∈ eval ρ (ex , ϕ)
      apply Not_Empty_iff_Contains_Elements in Hm.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m: M
Hm: ∃ x0 : M,
  x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
∃ x : M, x ∈ eval ρ (ex , ϕ)
      destruct Hm as [x0 Hx0].
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m, x0: M
Hx0: x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
∃ x : M, x ∈ eval ρ (ex , ϕ)
      exists x0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m, x0: M
Hx0: x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0 ∈ eval ρ (ex , ϕ)
      rewrite eval_ex_simpl.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m, x0: M
Hx0: x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0
∈ (let x := fresh_evar ϕ in
   propset_fa_union
     (λ e : M,
        let ρ' := update_evar_val x e ρ in
        eval ρ' ϕ^{evar:0↦x})) simpl.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m, x0: M
Hx0: x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0
∈ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      unfold propset_fa_union.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m, x0: M
Hx0: x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0
∈ {[ x | ∃ c : M,
           x
           ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
               ϕ^{evar:0↦fresh_evar ϕ} ]} rewrite -> elem_of_PropSet.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m, x0: M
Hx0: x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
∃ c : M,
  x0
  ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
      ϕ^{evar:0↦fresh_evar ϕ}
      exists m.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
Hpred: M_predicate ϕ^{evar:0↦x}
Hpredex: M_predicate (ex , ϕ)
m, x0: M
Hx0: x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0
∈ eval (update_evar_val (fresh_evar ϕ) m ρ)
    ϕ^{evar:0↦fresh_evar ϕ} assumption.
  Qed.

  Lemma eval_exists_empty ϕ ρ :
    let x := fresh_evar ϕ in
    eval ρ (patt_exists ϕ) = ∅ <->
    ∀ (m : Domain M), eval (update_evar_val x m ρ) (ϕ^{evar: 0 ↦ x}) = ∅.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
let x := fresh_evar ϕ in
eval ρ (ex , ϕ) = ∅
↔ (∀ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅)
  Proof.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
let x := fresh_evar ϕ in
eval ρ (ex , ϕ) = ∅
↔ (∀ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅)
    intros x.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
eval ρ (ex , ϕ) = ∅
↔ (∀ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅)
    rewrite -> eval_ex_simpl.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
(let x := fresh_evar ϕ in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ' ϕ^{evar:0↦x})) = ∅
↔ (∀ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅) simpl.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) = ∅
↔ (∀ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅)
    rewrite -> set_eq_subseteq.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
↔ (∀ m : M,
     eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅)
    split.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
→ ∀ m : M,
    eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
(∀ m : M,
   eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅)
→ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
  ⊆ propset_fa_union
      (λ e : M,
         eval (update_evar_val (fresh_evar ϕ) e ρ)
           ϕ^{evar:0↦fresh_evar ϕ})
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
→ ∀ m : M,
    eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅ intros [H1 _] m.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
m: M
eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
      rewrite -> elem_of_subseteq in H1.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  x
  ∈ propset_fa_union
      (λ e : M,
         eval
           (update_evar_val (fresh_evar ϕ) e ρ)
           ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
      unfold propset_fa_union in H1.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  x
  ∈ {[ x0 | ∃ c : M,
              x0
              ∈ eval
                  (update_evar_val 
                     (fresh_evar ϕ) c ρ)
                  ϕ^{evar:0↦
                  fresh_evar ϕ} ]} → 
  x ∈ ∅
m: M
eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
      setoid_rewrite -> elem_of_PropSet in H1.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
      rewrite -> set_eq_subseteq.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
eval (update_evar_val x m ρ) ϕ^{evar:0↦x} ⊆ ∅
⊆ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
      split.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
eval (update_evar_val x m ρ) ϕ^{evar:0↦x} ⊆ ∅
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
∅ ⊆ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
      2: {
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
∅ ⊆ eval (update_evar_val x m ρ) ϕ^{evar:0↦x} apply empty_subseteq. }
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
eval (update_evar_val x m ρ) ϕ^{evar:0↦x} ⊆ ∅
      rewrite -> elem_of_subseteq.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m: M
∀ x0 : M,
  x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
  → x0 ∈ ∅
      intros x0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m, x0: M
x0 ∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
→ x0 ∈ ∅
      intros Contra.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H1: ∀ x : M,
  (∃ c : M,
     x
     ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → 
  x ∈ ∅
m, x0: M
Contra: x0
∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0 ∈ ∅
      specialize (H1 x0).
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
x0: M
H1: (∃ c : M,
   x0
   ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) → 
x0 ∈ ∅
m: M
Contra: x0
∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0 ∈ ∅ apply H1.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
x0: M
H1: (∃ c : M,
   x0
   ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) → 
x0 ∈ ∅
m: M
Contra: x0
∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
∃ c : M,
  x0
  ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
      ϕ^{evar:0↦fresh_evar ϕ}
      exists m.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
x0: M
H1: (∃ c : M,
   x0
   ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) → 
x0 ∈ ∅
m: M
Contra: x0
∈ eval (update_evar_val x m ρ) ϕ^{evar:0↦x}
x0
∈ eval (update_evar_val (fresh_evar ϕ) m ρ)
    ϕ^{evar:0↦fresh_evar ϕ} assumption.
    -
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
(∀ m : M,
   eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅)
→ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
  ⊆ propset_fa_union
      (λ e : M,
         eval (update_evar_val (fresh_evar ϕ) e ρ)
           ϕ^{evar:0↦fresh_evar ϕ}) intros H.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      split.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
      2: {
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
∅
⊆ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) apply empty_subseteq. }
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ) e ρ)
       ϕ^{evar:0↦fresh_evar ϕ}) ⊆ ∅
      rewrite -> elem_of_subseteq.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
∀ x : M,
  x
  ∈ propset_fa_union
      (λ e : M,
         eval (update_evar_val (fresh_evar ϕ) e ρ)
           ϕ^{evar:0↦fresh_evar ϕ}) → x ∈ ∅
      intros x0.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
x0: M
x0
∈ propset_fa_union
    (λ e : M,
       eval (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ}) → x0 ∈ ∅
      intros Contra.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
x0: M
Contra: x0
∈ propset_fa_union
    (λ e : M,
       eval
         (update_evar_val (fresh_evar ϕ) e ρ)
         ϕ^{evar:0↦fresh_evar ϕ})
x0 ∈ ∅
      unfold propset_fa_union in Contra.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
x0: M
Contra: x0
∈ {[ x | ∃ c : M,
           x
           ∈ eval
               (update_evar_val
                  (fresh_evar ϕ) c ρ)
               ϕ^{evar:0↦
               fresh_evar ϕ} ]}
x0 ∈ ∅
      rewrite -> elem_of_PropSet in Contra.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
x0: M
Contra: ∃ c : M,
  x0
  ∈ eval (update_evar_val (fresh_evar ϕ) c ρ)
      ϕ^{evar:0↦fresh_evar ϕ}
x0 ∈ ∅
      destruct Contra as [m Contra].
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
x0, m: M
Contra: x0
∈ eval (update_evar_val (fresh_evar ϕ) m ρ)
    ϕ^{evar:0↦fresh_evar ϕ}
x0 ∈ ∅
      rewrite H in Contra.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
x:= fresh_evar ϕ: evar
H: ∀ m : M,
  eval (update_evar_val x m ρ) ϕ^{evar:0↦x} = ∅
x0, m: M
Contra: x0 ∈ ∅
x0 ∈ ∅
      exact Contra.
  Qed.


  (* Theory,axiom ref. snapshot: Definition 5 *)
  End with_explicit_model.

  Definition satisfies_model (M : Model) (ϕ : Pattern) : Prop :=
    forall (ρ : Valuation),
      eval (M := M) ρ ϕ = ⊤.

  Definition satisfies_theory (m : Model) (theory : Theory)
    : Prop := forall axiom : Pattern, axiom ∈ theory -> (satisfies_model m axiom).

  (* TODO rename *)
  Definition satisfies (theory : Theory) (p: Pattern)
    : Prop := forall m : Model, (satisfies_theory m theory) -> (satisfies_model m p).

  Lemma satisfies_theory_iff_satisfies_named_axioms {Σ : Signature} NAs M:
    satisfies_theory M (theory_of_NamedAxioms NAs) <->
    forall (n : NAName NAs), satisfies_model M (NAAxiom _ n).
signature, Σ: Signature
NAs: NamedAxioms
M: Model
satisfies_theory M (theory_of_NamedAxioms NAs)
↔ (∀ n : NAName NAs, satisfies_model M (NAAxiom NAs n))
  Proof.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
satisfies_theory M (theory_of_NamedAxioms NAs)
↔ (∀ n : NAName NAs, satisfies_model M (NAAxiom NAs n))
    split.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
satisfies_theory M (theory_of_NamedAxioms NAs)
→ ∀ n : NAName NAs, satisfies_model M (NAAxiom NAs n)
signature, Σ: Signature
NAs: NamedAxioms
M: Model
(∀ n : NAName NAs, satisfies_model M (NAAxiom NAs n))
→ satisfies_theory M (theory_of_NamedAxioms NAs)
    -
signature, Σ: Signature
NAs: NamedAxioms
M: Model
satisfies_theory M (theory_of_NamedAxioms NAs)
→ ∀ n : NAName NAs, satisfies_model M (NAAxiom NAs n) intros.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: satisfies_theory M (theory_of_NamedAxioms NAs)
n: NAName NAs
satisfies_model M (NAAxiom NAs n) unfold satisfies_theory in H.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: ∀ axiom : Pattern,
  axiom ∈ theory_of_NamedAxioms NAs
  → satisfies_model M axiom
n: NAName NAs
satisfies_model M (NAAxiom NAs n)
      specialize (H (NAAxiom _ n)).
signature, Σ: Signature
NAs: NamedAxioms
M: Model
n: NAName NAs
H: NAAxiom NAs n ∈ theory_of_NamedAxioms NAs
→ satisfies_model M (NAAxiom NAs n)
satisfies_model M (NAAxiom NAs n) apply H.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
n: NAName NAs
H: NAAxiom NAs n ∈ theory_of_NamedAxioms NAs
→ satisfies_model M (NAAxiom NAs n)
NAAxiom NAs n ∈ theory_of_NamedAxioms NAs
      unfold In.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
n: NAName NAs
H: NAAxiom NAs n ∈ theory_of_NamedAxioms NAs
→ satisfies_model M (NAAxiom NAs n)
NAAxiom NAs n ∈ theory_of_NamedAxioms NAs unfold theory_of_NamedAxioms.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
n: NAName NAs
H: NAAxiom NAs n ∈ theory_of_NamedAxioms NAs
→ satisfies_model M (NAAxiom NAs n)
NAAxiom NAs n
∈ {[ p | ∃ n : NAName NAs, p = NAAxiom NAs n ]}
      exists n.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
n: NAName NAs
H: NAAxiom NAs n ∈ theory_of_NamedAxioms NAs
→ satisfies_model M (NAAxiom NAs n)
NAAxiom NAs n = NAAxiom NAs n auto.
    -
signature, Σ: Signature
NAs: NamedAxioms
M: Model
(∀ n : NAName NAs, satisfies_model M (NAAxiom NAs n))
→ satisfies_theory M (theory_of_NamedAxioms NAs) intros H.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: ∀ n : NAName NAs,
  satisfies_model M (NAAxiom NAs n)
satisfies_theory M (theory_of_NamedAxioms NAs)
      intros ax Hax.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: ∀ n : NAName NAs,
  satisfies_model M (NAAxiom NAs n)
ax: Pattern
Hax: ax ∈ theory_of_NamedAxioms NAs
satisfies_model M ax
      unfold In in Hax.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: ∀ n : NAName NAs,
  satisfies_model M (NAAxiom NAs n)
ax: Pattern
Hax: ax ∈ theory_of_NamedAxioms NAs
satisfies_model M ax
      unfold theory_of_NamedAxioms in Hax.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: ∀ n : NAName NAs,
  satisfies_model M (NAAxiom NAs n)
ax: Pattern
Hax: ax
∈ {[ p | ∃ n : NAName NAs, p = NAAxiom NAs n ]}
satisfies_model M ax
      destruct Hax as [axname Haxname].
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: ∀ n : NAName NAs,
  satisfies_model M (NAAxiom NAs n)
ax: Pattern
axname: NAName NAs
Haxname: ax = NAAxiom NAs axname
satisfies_model M ax
      subst ax.
signature, Σ: Signature
NAs: NamedAxioms
M: Model
H: ∀ n : NAName NAs,
  satisfies_model M (NAAxiom NAs n)
axname: NAName NAs
satisfies_model M (NAAxiom NAs axname)
      apply H.
  Qed.

  Lemma satisfies_theory_subseteq M Γ₁ Γ₂:
    Γ₁ ⊆ Γ₂ ->
    satisfies_theory M Γ₂ ->
    satisfies_theory M Γ₁.
signature: Signature
M: Model
Γ₁, Γ₂: Theory
Γ₁ ⊆ Γ₂
→ satisfies_theory M Γ₂ → satisfies_theory M Γ₁
  Proof.
signature: Signature
M: Model
Γ₁, Γ₂: Theory
Γ₁ ⊆ Γ₂
→ satisfies_theory M Γ₂ → satisfies_theory M Γ₁
    unfold satisfies_theory.
signature: Signature
M: Model
Γ₁, Γ₂: Theory
Γ₁ ⊆ Γ₂
→ (∀ axiom : Pattern,
     axiom ∈ Γ₂ → satisfies_model M axiom)
  → ∀ axiom : Pattern,
      axiom ∈ Γ₁ → satisfies_model M axiom
    set_solver by fail.
  Qed.


  (* theory predicate *)
  Definition T_predicate Γ ϕ := forall M, satisfies_theory M Γ -> M_predicate M ϕ.

  Hint Extern 4 (M_predicate _ (evar_open _ _ _)) => mlSimpl : core.
  Hint Extern 4 (T_predicate _ (evar_open _ _ _)) => mlSimpl : core.
  Hint Extern 4 (M_predicate _ (svar_open _ _ _)) => mlSimpl : core.
  Hint Extern 4 (T_predicate _ (svar_open _ _ _)) => mlSimpl : core.

  Lemma T_predicate_impl_M_predicate M Γ ϕ:
    satisfies_theory M Γ -> T_predicate Γ ϕ -> M_predicate M ϕ.
signature: Signature
M: Model
Γ: Theory
ϕ: Pattern
satisfies_theory M Γ
→ T_predicate Γ ϕ → M_predicate M ϕ
  Proof.
signature: Signature
M: Model
Γ: Theory
ϕ: Pattern
satisfies_theory M Γ
→ T_predicate Γ ϕ → M_predicate M ϕ
    unfold T_predicate.
signature: Signature
M: Model
Γ: Theory
ϕ: Pattern
satisfies_theory M Γ
→ (∀ M : Model, satisfies_theory M Γ → M_predicate M ϕ)
  → M_predicate M ϕ auto.
  Qed.

  Hint Resolve T_predicate_impl_M_predicate : core.

  Lemma T_predicate_impl Γ ϕ₁ ϕ₂ : T_predicate Γ ϕ₁ -> T_predicate Γ ϕ₂ -> T_predicate Γ (patt_imp ϕ₁ ϕ₂).
signature: Signature
Γ: Theory
ϕ₁, ϕ₂: Pattern
T_predicate Γ ϕ₁
→ T_predicate Γ ϕ₂ → T_predicate Γ (ϕ₁ ---> ϕ₂)
  Proof.
signature: Signature
Γ: Theory
ϕ₁, ϕ₂: Pattern
T_predicate Γ ϕ₁
→ T_predicate Γ ϕ₂ → T_predicate Γ (ϕ₁ ---> ϕ₂)
    unfold T_predicate.
signature: Signature
Γ: Theory
ϕ₁, ϕ₂: Pattern
(∀ M : Model, satisfies_theory M Γ → M_predicate M ϕ₁)
→ (∀ M : Model,
     satisfies_theory M Γ → M_predicate M ϕ₂)
  → ∀ M : Model,
      satisfies_theory M Γ
      → M_predicate M (ϕ₁ ---> ϕ₂)
    intros.
signature: Signature
Γ: Theory
ϕ₁, ϕ₂: Pattern
H: ∀ M : Model,
  satisfies_theory M Γ → M_predicate M ϕ₁
H0: ∀ M : Model,
  satisfies_theory M Γ → M_predicate M ϕ₂
M: Model
H1: satisfies_theory M Γ
M_predicate M (ϕ₁ ---> ϕ₂)
    auto using M_predicate_impl.
  Qed.

  Hint Resolve T_predicate_impl : core.

  Lemma T_predicate_bot Γ : T_predicate Γ patt_bott.
signature: Signature
Γ: Theory
T_predicate Γ ⊥
  Proof.
signature: Signature
Γ: Theory
T_predicate Γ ⊥
    unfold T_predicate.
signature: Signature
Γ: Theory
∀ M : Model, satisfies_theory M Γ → M_predicate M ⊥
    intros.
signature: Signature
Γ: Theory
M: Model
H: satisfies_theory M Γ
M_predicate M ⊥
    auto using M_predicate_bott.
  Qed.

  Hint Resolve T_predicate_bot : core.

  (* TODO: top iff exists forall *)

Section with_model.
  Context {M : Model}.

  (* If phi1 \subseteq phi2, then U_x phi1 \subseteq U_x phi2 *)
  Lemma eval_subset_union x ϕ₁ ϕ₂ :
    (forall ρ, (eval ρ ϕ₁) ⊆ (@eval M ρ ϕ₂) )
    -> (forall ρ,
                    (propset_fa_union (fun e => eval (update_evar_val x e ρ) ϕ₁))
                    ⊆
                    (propset_fa_union (fun e => @eval M (update_evar_val x e ρ) ϕ₂))
       ).
signature: Signature
M: Model
x: evar
ϕ₁, ϕ₂: Pattern
(∀ ρ : Valuation, eval ρ ϕ₁ ⊆ eval ρ ϕ₂)
→ ∀ ρ : Valuation,
    propset_fa_union
      (λ e : M, eval (update_evar_val x e ρ) ϕ₁)
    ⊆ propset_fa_union
        (λ e : M, eval (update_evar_val x e ρ) ϕ₂)
  Proof.
signature: Signature
M: Model
x: evar
ϕ₁, ϕ₂: Pattern
(∀ ρ : Valuation, eval ρ ϕ₁ ⊆ eval ρ ϕ₂)
→ ∀ ρ : Valuation,
    propset_fa_union
      (λ e : M, eval (update_evar_val x e ρ) ϕ₁)
    ⊆ propset_fa_union
        (λ e : M, eval (update_evar_val x e ρ) ϕ₂)
    intros H.
signature: Signature
M: Model
x: evar
ϕ₁, ϕ₂: Pattern
H: ∀ ρ : Valuation, eval ρ ϕ₁ ⊆ eval ρ ϕ₂
∀ ρ : Valuation,
  propset_fa_union
    (λ e : M, eval (update_evar_val x e ρ) ϕ₁)
  ⊆ propset_fa_union
      (λ e : M, eval (update_evar_val x e ρ) ϕ₂) induction ϕ₁; intros; apply propset_fa_union_included; auto.
  Qed.

  (* eval unchanged when using fresh element varaiable *)
  Lemma eval_free_evar_independent ρ x v ϕ:
    evar_is_fresh_in x ϕ ->
    @eval M (update_evar_val x v ρ) ϕ = eval ρ ϕ.
signature: Signature
M: Model
ρ: Valuation
x: evar
v: M
ϕ: Pattern
evar_is_fresh_in x ϕ
→ eval (update_evar_val x v ρ) ϕ = eval ρ ϕ
  Proof.
signature: Signature
M: Model
ρ: Valuation
x: evar
v: M
ϕ: Pattern
evar_is_fresh_in x ϕ
→ eval (update_evar_val x v ρ) ϕ = eval ρ ϕ
    intros Hfr.
signature: Signature
M: Model
ρ: Valuation
x: evar
v: M
ϕ: Pattern
Hfr: evar_is_fresh_in x ϕ
eval (update_evar_val x v ρ) ϕ = eval ρ ϕ unfold evar_is_fresh_in in Hfr.
signature: Signature
M: Model
ρ: Valuation
x: evar
v: M
ϕ: Pattern
Hfr: x ∉ free_evars ϕ
eval (update_evar_val x v ρ) ϕ = eval ρ ϕ
    funelim (eval (update_evar_val x v ρ) ϕ); try simp eval; try reflexivity;
      simpl in Hfr.
signature: Signature
M: Model
ρ0: Valuation
x0: evar
v: M
x: evar
Hfr: x0 ∉ {[x]}
{[evar_valuation (update_evar_val x0 v ρ0) x]} =
{[evar_valuation ρ0 x]}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
(let x0 := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x0 e (update_evar_val x v ρ0)
        in
      eval ρ' ϕ'^{evar:0↦x0})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ0 in
      eval ρ' ϕ'^{evar:0↦x}))
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' :=
        update_svar_val X S (update_evar_val x v ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X}))
    -
signature: Signature
M: Model
ρ0: Valuation
x0: evar
v: M
x: evar
Hfr: x0 ∉ {[x]}
{[evar_valuation (update_evar_val x0 v ρ0) x]} =
{[evar_valuation ρ0 x]} simpl.
signature: Signature
M: Model
ρ0: Valuation
x0: evar
v: M
x: evar
Hfr: x0 ∉ {[x]}
{[if decide (x0 = x) then v else evar_valuation ρ0 x]} =
{[evar_valuation ρ0 x]} destruct (decide (x0 = x)); simpl.
signature: Signature
M: Model
ρ0: Valuation
x0: evar
v: M
x: evar
Hfr: x0 ∉ {[x]}
e: x0 = x
{[v]} = {[evar_valuation ρ0 x]}
signature: Signature
M: Model
ρ0: Valuation
x0: evar
v: M
x: evar
Hfr: x0 ∉ {[x]}
n: x0 ≠ x
{[evar_valuation ρ0 x]} = {[evar_valuation ρ0 x]}
      +
signature: Signature
M: Model
ρ0: Valuation
x0: evar
v: M
x: evar
Hfr: x0 ∉ {[x]}
e: x0 = x
{[v]} = {[evar_valuation ρ0 x]} subst.
signature: Signature
M: Model
ρ0: Valuation
v: M
x: evar
Hfr: x ∉ {[x]}
{[v]} = {[evar_valuation ρ0 x]} unfold evar_is_fresh_in in Hfr.
signature: Signature
M: Model
ρ0: Valuation
v: M
x: evar
Hfr: x ∉ {[x]}
{[v]} = {[evar_valuation ρ0 x]} simpl in Hfr.
signature: Signature
M: Model
ρ0: Valuation
v: M
x: evar
Hfr: x ∉ {[x]}
{[v]} = {[evar_valuation ρ0 x]}
        apply not_elem_of_singleton_1 in Hfr.
signature: Signature
M: Model
ρ0: Valuation
v: M
x: evar
Hfr: x ≠ x
{[v]} = {[evar_valuation ρ0 x]}
        contradiction.
      +
signature: Signature
M: Model
ρ0: Valuation
x0: evar
v: M
x: evar
Hfr: x0 ∉ {[x]}
n: x0 ≠ x
{[evar_valuation ρ0 x]} = {[evar_valuation ρ0 x]} reflexivity.
    -
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂) rewrite H.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
x ∉ free_evars ϕ₁
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
x ∉ free_evars ϕ₁ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      rewrite H0.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
x ∉ free_evars ϕ₂
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
x ∉ free_evars ϕ₂ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: app_ext (eval (update_evar_val x v ρ0) ϕ₁)
  (eval (update_evar_val x v ρ0) ϕ₂) =
eval (update_evar_val x v ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      reflexivity.
    -
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂ rewrite H.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
x ∉ free_evars ϕ₁
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval (update_evar_val x v ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
x ∉ free_evars ϕ₁ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval (update_evar_val x v ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      rewrite H0.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
x ∉ free_evars ϕ₂
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
x ∉ free_evars ϕ₂ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ₁, ϕ₂: Pattern
H: x ∉ free_evars ϕ₁
→ eval (update_evar_val x v ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: x ∉ free_evars ϕ₂
→ eval (update_evar_val x v ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: x ∉ free_evars ϕ₁ ∪ free_evars ϕ₂
Heqcall: ⊤ ∖ eval (update_evar_val x v ρ0) ϕ₁
∪ eval (update_evar_val x v ρ0) ϕ₂ =
eval (update_evar_val x v ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      reflexivity.
    -
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
(let x0 := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x0 e (update_evar_val x v ρ0)
        in
      eval ρ' ϕ'^{evar:0↦x0})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ0 in
      eval ρ' ϕ'^{evar:0↦x})) simpl.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ') e ρ0)
       ϕ'^{evar:0↦fresh_evar ϕ'})
      apply f_equal.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
(λ e : M,
   eval
     (update_evar_val (fresh_evar ϕ') e
        (update_evar_val x v ρ0))
     ϕ'^{evar:0↦fresh_evar ϕ'}) =
(λ e : M,
   eval (update_evar_val (fresh_evar ϕ') e ρ0)
     ϕ'^{evar:0↦fresh_evar ϕ'}) apply functional_extensionality.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
∀ x0 : M,
  eval
    (update_evar_val (fresh_evar ϕ') x0
       (update_evar_val x v ρ0))
    ϕ'^{evar:0↦fresh_evar ϕ'} =
  eval (update_evar_val (fresh_evar ϕ') x0 ρ0)
    ϕ'^{evar:0↦fresh_evar ϕ'} intros e.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
      destruct (decide ((fresh_evar ϕ') = x)) as [Heq|Hneq].
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Heq: fresh_evar ϕ' = x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
      +
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Heq: fresh_evar ϕ' = x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} rewrite -> Heq.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Heq: fresh_evar ϕ' = x
eval (update_evar_val x e (update_evar_val x v ρ0))
  ϕ'^{evar:0↦x} =
eval (update_evar_val x e ρ0) ϕ'^{evar:0↦x} rewrite -> update_evar_val_shadow.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Heq: fresh_evar ϕ' = x
eval (update_evar_val x e ρ0) ϕ'^{evar:0↦x} =
eval (update_evar_val x e ρ0) ϕ'^{evar:0↦x} reflexivity.
      +
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} rewrite -> update_evar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val x v
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
fresh_evar ϕ' ≠ x
        2: apply Hneq.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val x v
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
        rewrite -> H with (e := e).
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
x ∉ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_evar_val x v ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_evar_val x v
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
        {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} reflexivity. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
x ∉ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_evar_val x v ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_evar_val x v
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
        {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
x ∉ free_evars ϕ'^{evar:0↦fresh_evar ϕ'} 
          intros Contra.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
False
          pose proof (Hfeeo := @free_evars_evar_open signature ϕ' (fresh_evar ϕ') 0).
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
Hfeeo: free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
⊆ {[fresh_evar ϕ']} ∪ free_evars ϕ'
False
           rewrite -> elem_of_subseteq in Hfeeo.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
Hfeeo: ∀ x : evar,
  x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
  → x ∈ {[fresh_evar ϕ']} ∪ free_evars ϕ'
False
           specialize (Hfeeo x Contra).
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
Hfeeo: x ∈ {[fresh_evar ϕ']} ∪ free_evars ϕ'
False
           apply elem_of_union in Hfeeo.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
Hfeeo: x ∈ {[fresh_evar ϕ']} ∨ x ∈ free_evars ϕ'
False
           destruct Hfeeo as [H'|H'].
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
H': x ∈ {[fresh_evar ϕ']}
False
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
H': x ∈ free_evars ϕ'
False
           *
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
H': x ∈ {[fresh_evar ϕ']}
False apply elem_of_singleton_1 in H'.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
H': x = fresh_evar ϕ'
False symmetry in H'.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
H': fresh_evar ϕ' = x
False contradiction.
           *
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
Contra: x ∈ free_evars ϕ'^{evar:0↦fresh_evar ϕ'}
H': x ∈ free_evars ϕ'
False contradiction.
      }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_evar_val x v ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_evar_val x v
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_evar_val x v ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} f_equal.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
update_evar_val (fresh_evar ϕ') e
  (update_evar_val x v ρ0) =
update_evar_val x v
  (update_evar_val (fresh_evar ϕ') e ρ0) apply update_evar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
fresh_evar ϕ' ≠ x apply Hneq. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_evar_val x v
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_evar_val x v
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} f_equal.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
update_evar_val (fresh_evar ϕ') e
  (update_evar_val x v ρ0) =
update_evar_val x v
  (update_evar_val (fresh_evar ϕ') e ρ0) apply update_evar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_evar_val x v ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_evar_val x v ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_evar_val x v ρ0) (ex , ϕ')
e: M
Hneq: fresh_evar ϕ' ≠ x
fresh_evar ϕ' ≠ x apply Hneq. }
    -
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' :=
        update_svar_val X S (update_evar_val x v ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X})) simpl.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
LeastFixpointOf
  (λ S : propset M,
     eval (update_svar_val (fresh_svar ϕ') S ρ0)
       ϕ'^{svar:0↦fresh_svar ϕ'})
      apply f_equal.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
(λ S : propset M,
   eval
     (update_svar_val (fresh_svar ϕ') S
        (update_evar_val x v ρ0))
     ϕ'^{svar:0↦fresh_svar ϕ'}) =
(λ S : propset M,
   eval (update_svar_val (fresh_svar ϕ') S ρ0)
     ϕ'^{svar:0↦fresh_svar ϕ'}) apply functional_extensionality.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
∀ x0 : propset M,
  eval
    (update_svar_val (fresh_svar ϕ') x0
       (update_evar_val x v ρ0))
    ϕ'^{svar:0↦fresh_svar ϕ'} =
  eval (update_svar_val (fresh_svar ϕ') x0 ρ0)
    ϕ'^{svar:0↦fresh_svar ϕ'}
      intros e.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
      rewrite -update_evar_val_svar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval
  (update_evar_val x v
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
      erewrite -> H.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
x ∉ free_evars ϕ'^{svar:0↦fresh_svar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') 
      ?S (update_evar_val x v ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval
  (update_svar_val (fresh_svar ϕ') 
     ?S (update_evar_val x v ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_evar_val x v
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} reflexivity. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
x ∉ free_evars ϕ'^{svar:0↦fresh_svar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') 
      ?S (update_evar_val x v ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval
  (update_svar_val (fresh_svar ϕ') 
     ?S (update_evar_val x v ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_evar_val x v
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
x ∉ free_evars ϕ'^{svar:0↦fresh_svar ϕ'} intros Contra.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
Contra: x ∈ free_evars ϕ'^{svar:0↦fresh_svar ϕ'}
False
         rewrite -> free_evars_svar_open in Contra.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
Contra: x ∈ free_evars ϕ'
False
         contradiction.
      }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') ?S
      (update_evar_val x v ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval
  (update_svar_val (fresh_svar ϕ') 
     ?S (update_evar_val x v ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_evar_val x v
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') ?S
      (update_evar_val x v ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_evar_val x v
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} reflexivity. }
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_evar_val x v
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
      rewrite update_evar_val_svar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
x: evar
v: M
ϕ': Pattern
H: ∀ (S : propset M) (ρ : Valuation) 
  (x0 : evar) (v0 : M) 
  (ϕ : Pattern),
  x0 ∉ free_evars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_evar_val x0 v0 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S
           (update_evar_val x v ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_evar_val x0 v0 ρ) ϕ
      → eval (update_evar_val x0 v0 ρ) ϕ =
        eval ρ ϕ
Hfr: x ∉ free_evars ϕ'
Heqcall: LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S
          (update_evar_val x v ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_evar_val x v ρ0) (mu , ϕ')
e: propset M
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_evar_val x v ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
      reflexivity.
  Qed.

  Lemma eval_free_svar_independent ρ X S ϕ:
    svar_is_fresh_in X ϕ ->
    @eval M (update_svar_val X S ρ) ϕ = eval ρ ϕ.
signature: Signature
M: Model
ρ: Valuation
X: svar
S: propset M
ϕ: Pattern
svar_is_fresh_in X ϕ
→ eval (update_svar_val X S ρ) ϕ = eval ρ ϕ
  Proof.
signature: Signature
M: Model
ρ: Valuation
X: svar
S: propset M
ϕ: Pattern
svar_is_fresh_in X ϕ
→ eval (update_svar_val X S ρ) ϕ = eval ρ ϕ
    intros Hfr.
signature: Signature
M: Model
ρ: Valuation
X: svar
S: propset M
ϕ: Pattern
Hfr: svar_is_fresh_in X ϕ
eval (update_svar_val X S ρ) ϕ = eval ρ ϕ unfold svar_is_fresh_in in Hfr.
signature: Signature
M: Model
ρ: Valuation
X: svar
S: propset M
ϕ: Pattern
Hfr: X ∉ free_svars ϕ
eval (update_svar_val X S ρ) ϕ = eval ρ ϕ
    funelim (eval (update_svar_val X S ρ) ϕ);
      try simp eval; try reflexivity;
      simpl in Hfr.
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
svar_valuation (update_svar_val X0 S ρ0) X =
svar_valuation ρ0 X
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{evar:0↦x})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ0 in
      eval ρ' ϕ'^{evar:0↦x}))
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
(let X0 := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0 (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X0})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X}))
    {
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
svar_valuation (update_svar_val X0 S ρ0) X =
svar_valuation ρ0 X
      unfold update_svar_val.
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
svar_valuation
  {|
    evar_valuation := evar_valuation ρ0;
    svar_valuation :=
      λ sv' : svar,
        if decide (X0 = sv')
        then S
        else svar_valuation ρ0 sv'
  |} X = svar_valuation ρ0 X simpl.
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
(if decide (X0 = X) then S else svar_valuation ρ0 X) =
svar_valuation ρ0 X
      destruct (decide (X0 = X)); simpl.
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
e: X0 = X
S = svar_valuation ρ0 X
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
n: X0 ≠ X
svar_valuation ρ0 X = svar_valuation ρ0 X
      +
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
e: X0 = X
S = svar_valuation ρ0 X subst.
signature: Signature
M: Model
ρ0: Valuation
S: propset M
X: svar
Hfr: X ∉ {[X]}
S = svar_valuation ρ0 X
        destruct ρ0 as [ρₑ ρₛ].
signature: Signature
M: Model
ρₑ: evar → M
ρₛ: svar → propset M
S: propset M
X: svar
Hfr: X ∉ {[X]}
S =
svar_valuation
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} X simpl.
signature: Signature
M: Model
ρₑ: evar → M
ρₛ: svar → propset M
S: propset M
X: svar
Hfr: X ∉ {[X]}
S = ρₛ X
        apply not_elem_of_singleton_1 in Hfr.
signature: Signature
M: Model
ρₑ: evar → M
ρₛ: svar → propset M
S: propset M
X: svar
Hfr: X ≠ X
S = ρₛ X
        contradiction.
      +
signature: Signature
M: Model
ρ0: Valuation
X0: svar
S: propset M
X: svar
Hfr: X0 ∉ {[X]}
n: X0 ≠ X
svar_valuation ρ0 X = svar_valuation ρ0 X reflexivity.
    }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{evar:0↦x})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ0 in
      eval ρ' ϕ'^{evar:0↦x}))
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
(let X0 := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0 (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X0})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X}))
    {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      rewrite H.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
X ∉ free_svars ϕ₁
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
X ∉ free_svars ϕ₁ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      rewrite H0.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
X ∉ free_svars ϕ₂
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
X ∉ free_svars ϕ₂ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: app_ext (eval (update_svar_val X S ρ0) ϕ₁)
  (eval (update_svar_val X S ρ0) ϕ₂) =
eval (update_svar_val X S ρ0) (ϕ₁ $ ϕ₂)
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂) =
app_ext (eval ρ0 ϕ₁) (eval ρ0 ϕ₂)
      reflexivity.
    }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{evar:0↦x})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ0 in
      eval ρ' ϕ'^{evar:0↦x}))
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
(let X0 := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0 (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X0})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X}))
    {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      rewrite H.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
X ∉ free_svars ϕ₁
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval (update_svar_val X S ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
X ∉ free_svars ϕ₁ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval (update_svar_val X S ρ0) ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      rewrite H0.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
X ∉ free_svars ϕ₂
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
X ∉ free_svars ϕ₂ set_solver. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ₁, ϕ₂: Pattern
H: X ∉ free_svars ϕ₁
→ eval (update_svar_val X S ρ0) ϕ₁ = eval ρ0 ϕ₁
H0: X ∉ free_svars ϕ₂
→ eval (update_svar_val X S ρ0) ϕ₂ = eval ρ0 ϕ₂
Hfr: X ∉ free_svars ϕ₁ ∪ free_svars ϕ₂
Heqcall: ⊤ ∖ eval (update_svar_val X S ρ0) ϕ₁
∪ eval (update_svar_val X S ρ0) ϕ₂ =
eval (update_svar_val X S ρ0) (ϕ₁ ---> ϕ₂)
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂ =
⊤ ∖ eval ρ0 ϕ₁ ∪ eval ρ0 ϕ₂
      reflexivity.
    }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{evar:0↦x})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ0 in
      eval ρ' ϕ'^{evar:0↦x}))
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
(let X0 := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0 (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X0})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X}))
    {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{evar:0↦x})) =
(let x := fresh_evar ϕ' in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ0 in
      eval ρ' ϕ'^{evar:0↦x}))
      simpl.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
propset_fa_union
  (λ e : M,
     eval (update_evar_val (fresh_evar ϕ') e ρ0)
       ϕ'^{evar:0↦fresh_evar ϕ'})
      apply f_equal.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
(λ e : M,
   eval
     (update_evar_val (fresh_evar ϕ') e
        (update_svar_val X S ρ0))
     ϕ'^{evar:0↦fresh_evar ϕ'}) =
(λ e : M,
   eval (update_evar_val (fresh_evar ϕ') e ρ0)
     ϕ'^{evar:0↦fresh_evar ϕ'}) apply functional_extensionality.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
∀ x : M,
  eval
    (update_evar_val (fresh_evar ϕ') x
       (update_svar_val X S ρ0))
    ϕ'^{evar:0↦fresh_evar ϕ'} =
  eval (update_evar_val (fresh_evar ϕ') x ρ0)
    ϕ'^{evar:0↦fresh_evar ϕ'} intros e.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
      rewrite update_evar_val_svar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
      rewrite -> H with (e := e).
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
X ∉ free_svars ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval (update_evar_val (fresh_evar ϕ') e ρ0)
  ϕ'^{evar:0↦fresh_evar ϕ'} reflexivity. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
X ∉ free_svars ϕ'^{evar:0↦fresh_evar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
X ∉ free_svars ϕ'^{evar:0↦fresh_evar ϕ'} apply svar_fresh_evar_open.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
svar_is_fresh_in X ϕ' apply Hfr. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
{|
  pr1 :=
    update_evar_val (fresh_evar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_evar_val (fresh_evar ϕ') e ρ0);
  pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
|} reflexivity. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'}
      {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_evar_val (fresh_evar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} rewrite update_evar_val_svar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (e : M) (ρ : Valuation) 
  (X0 : svar) (S0 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{evar:0↦fresh_evar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S0 ρ; pr2 := ϕ |}
    → eval
        (update_evar_val 
           (fresh_evar ϕ') e
           (update_svar_val X S ρ0))
        ϕ'^{evar:0↦fresh_evar ϕ'} =
      eval (update_svar_val X0 S0 ρ) ϕ
      → eval (update_svar_val X0 S0 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: propset_fa_union
  (λ e : M,
     eval
       (update_evar_val 
          (fresh_evar ϕ') e
          (update_svar_val X S ρ0))
       ϕ'^{evar:0↦fresh_evar ϕ'}) =
eval (update_svar_val X S ρ0) (ex , ϕ')
e: M
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} =
eval
  (update_svar_val X S
     (update_evar_val (fresh_evar ϕ') e ρ0))
  ϕ'^{evar:0↦fresh_evar ϕ'} reflexivity. }
    }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
(let X0 := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0 (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X0})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X}))
    {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
(let X0 := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0 (update_svar_val X S ρ0)
        in
      eval ρ' ϕ'^{svar:0↦X0})) =
(let X := fresh_svar ϕ' in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ0 in
      eval ρ' ϕ'^{svar:0↦X}))
      apply f_equal.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
(λ S0 : propset M,
   let ρ' :=
     update_svar_val (fresh_svar ϕ') S0
       (update_svar_val X S ρ0) in
   eval ρ' ϕ'^{svar:0↦fresh_svar ϕ'}) =
(λ S : propset M,
   let ρ' := update_svar_val (fresh_svar ϕ') S ρ0 in
   eval ρ' ϕ'^{svar:0↦fresh_svar ϕ'}) apply functional_extensionality.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
∀ x : propset M,
  (let ρ' :=
     update_svar_val (fresh_svar ϕ') x
       (update_svar_val X S ρ0) in
   eval ρ' ϕ'^{svar:0↦fresh_svar ϕ'}) =
  (let ρ' := update_svar_val (fresh_svar ϕ') x ρ0 in
   eval ρ' ϕ'^{svar:0↦fresh_svar ϕ'})
      intros e.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
(let ρ' :=
   update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0) in
 eval ρ' ϕ'^{svar:0↦fresh_svar ϕ'}) =
(let ρ' := update_svar_val (fresh_svar ϕ') e ρ0 in
 eval ρ' ϕ'^{svar:0↦fresh_svar ϕ'}) simpl.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
      destruct (decide ((fresh_svar ϕ') = X)) as [Heq|Hneq].
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Heq: fresh_svar ϕ' = X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
      +
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Heq: fresh_svar ϕ' = X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} rewrite Heq.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Heq: fresh_svar ϕ' = X
eval (update_svar_val X e (update_svar_val X S ρ0))
  ϕ'^{svar:0↦X} =
eval (update_svar_val X e ρ0) ϕ'^{svar:0↦X} rewrite update_svar_val_shadow.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Heq: fresh_svar ϕ' = X
eval (update_svar_val X e ρ0) ϕ'^{svar:0↦X} =
eval (update_svar_val X e ρ0) ϕ'^{svar:0↦X} reflexivity.
      +
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} rewrite update_svar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
fresh_svar ϕ' ≠ X
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val X S
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
        {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
fresh_svar ϕ' ≠ X apply Hneq.  }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val X S
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
        rewrite -> H with (S := e).
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
X ∉ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_svar_val X S
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
        {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval (update_svar_val (fresh_svar ϕ') e ρ0)
  ϕ'^{svar:0↦fresh_svar ϕ'} reflexivity. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
X ∉ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_svar_val X S
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
        {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
X ∉ free_svars ϕ'^{svar:0↦fresh_svar ϕ'} intros Contra.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
False
          pose proof (Hfeeo := @free_svars_svar_open signature ϕ' (fresh_svar ϕ') 0).
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
Hfeeo: free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
⊆ {[fresh_svar ϕ']} ∪ free_svars ϕ'
False
          rewrite -> elem_of_subseteq in Hfeeo.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
Hfeeo: ∀ x : svar,
  x ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
  → x ∈ {[fresh_svar ϕ']} ∪ free_svars ϕ'
False
          specialize (Hfeeo X Contra).
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
Hfeeo: X ∈ {[fresh_svar ϕ']} ∪ free_svars ϕ'
False
          apply elem_of_union in Hfeeo.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
Hfeeo: X ∈ {[fresh_svar ϕ']} ∨ X ∈ free_svars ϕ'
False
          destruct Hfeeo as [H'|H'].
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
H': X ∈ {[fresh_svar ϕ']}
False
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
H': X ∈ free_svars ϕ'
False
          *
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
H': X ∈ {[fresh_svar ϕ']}
False apply elem_of_singleton_1 in H'.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
H': X = fresh_svar ϕ'
False symmetry in H'.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
H': fresh_svar ϕ' = X
False contradiction.
          *
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
Contra: X ∈ free_svars ϕ'^{svar:0↦fresh_svar ϕ'}
H': X ∈ free_svars ϕ'
False contradiction.
        }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|}
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_svar_val X S
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
        {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
{|
  pr1 :=
    update_svar_val (fresh_svar ϕ') e
      (update_svar_val X S ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} =
{|
  pr1 :=
    update_svar_val X S
      (update_svar_val (fresh_svar ϕ') e ρ0);
  pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
|} f_equal.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
update_svar_val (fresh_svar ϕ') e
  (update_svar_val X S ρ0) =
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) rewrite update_svar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
fresh_svar ϕ' ≠ X
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) =
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) apply Hneq.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) =
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) reflexivity. }
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_svar_val X S
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'}
        {
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
eval
  (update_svar_val (fresh_svar ϕ') e
     (update_svar_val X S ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} =
eval
  (update_svar_val X S
     (update_svar_val (fresh_svar ϕ') e ρ0))
  ϕ'^{svar:0↦fresh_svar ϕ'} f_equal.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
update_svar_val (fresh_svar ϕ') e
  (update_svar_val X S ρ0) =
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) rewrite update_svar_val_comm.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
fresh_svar ϕ' ≠ X
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) =
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) apply Hneq.
signature: Signature
M: Model
ρ0: Valuation
X: svar
S: propset M
ϕ': Pattern
H: ∀ (S0 : propset M) (ρ : Valuation) 
  (X0 : svar) (S1 : propset M) 
  (ϕ : Pattern),
  X0 ∉ free_svars ϕ
  → {|
      pr1 :=
        update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0);
      pr2 := ϕ'^{svar:0↦fresh_svar ϕ'}
    |} =
    {| pr1 := update_svar_val X0 S1 ρ; pr2 := ϕ |}
    → eval
        (update_svar_val 
           (fresh_svar ϕ') S0
           (update_svar_val X S ρ0))
        ϕ'^{svar:0↦fresh_svar ϕ'} =
      eval (update_svar_val X0 S1 ρ) ϕ
      → eval (update_svar_val X0 S1 ρ) ϕ =
        eval ρ ϕ
Hfr: X ∉ free_svars ϕ'
Heqcall: LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val 
          (fresh_svar ϕ') S0
          (update_svar_val X S ρ0))
       ϕ'^{svar:0↦fresh_svar ϕ'}) =
eval (update_svar_val X S ρ0) (mu , ϕ')
e: propset M
Hneq: fresh_svar ϕ' ≠ X
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) =
update_svar_val X S
  (update_svar_val (fresh_svar ϕ') e ρ0) reflexivity. }
    }
  Qed.

  (* Can change updated/opened fresh variable variable *)
  Lemma Private_eval_fresh_var_open sz ϕ dbi ρ:
    size ϕ <= sz ->
    (
      forall X Y S,
        svar_is_fresh_in X ϕ ->
        svar_is_fresh_in Y ϕ ->
        eval (update_svar_val X S ρ) (ϕ^{svar: dbi ↦ X})
        = @eval M (update_svar_val Y S ρ) (ϕ^{svar: dbi ↦ Y})
    ) /\
    (
      forall x y c,
        evar_is_fresh_in x ϕ ->
        evar_is_fresh_in y ϕ ->
        @eval M (update_evar_val x c ρ) (ϕ^{evar: dbi ↦ x})
        = eval (update_evar_val y c ρ) (ϕ^{evar: dbi ↦ y})
    )
  .
signature: Signature
M: Model
sz: nat
ϕ: Pattern
dbi: db_index
ρ: Valuation
size ϕ ≤ sz
→ (∀ (X Y : svar) (S : propset M),
     svar_is_fresh_in X ϕ
     → svar_is_fresh_in Y ϕ
       → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
         eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
  ∧ (∀ (x y : evar) (c : M),
       evar_is_fresh_in x ϕ
       → evar_is_fresh_in y ϕ
         → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
           eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y})
  Proof.
signature: Signature
M: Model
sz: nat
ϕ: Pattern
dbi: db_index
ρ: Valuation
size ϕ ≤ sz
→ (∀ (X Y : svar) (S : propset M),
     svar_is_fresh_in X ϕ
     → svar_is_fresh_in Y ϕ
       → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
         eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
  ∧ (∀ (x y : evar) (c : M),
       evar_is_fresh_in x ϕ
       → evar_is_fresh_in y ϕ
         → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
           eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y})
    move: ϕ dbi ρ.
signature: Signature
M: Model
sz: nat
∀ (ϕ : Pattern) (dbi : db_index) (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
    induction sz.
signature: Signature
M: Model
∀ (ϕ : Pattern) (dbi : db_index) (ρ : Valuation),
  size ϕ ≤ 0
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
∀ (ϕ : Pattern) (dbi : db_index) (ρ : Valuation),
  size ϕ ≤ S sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
    -
signature: Signature
M: Model
∀ (ϕ : Pattern) (dbi : db_index) (ρ : Valuation),
  size ϕ ≤ 0
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y}) move=> ϕ dbi ρ Hsz.
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
(∀ (X Y : svar) (S : propset M),
   svar_is_fresh_in X ϕ
   → svar_is_fresh_in Y ϕ
     → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
       eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
∧ (∀ (x y : evar) (c : M),
     evar_is_fresh_in x ϕ
     → evar_is_fresh_in y ϕ
       → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
         eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y})
      split.
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
∀ (X Y : svar) (S : propset M),
  svar_is_fresh_in X ϕ
  → svar_is_fresh_in Y ϕ
    → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
      eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
∀ (x y : evar) (c : M),
  evar_is_fresh_in x ϕ
  → evar_is_fresh_in y ϕ
    → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
      eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
      {
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
∀ (X Y : svar) (S : propset M),
  svar_is_fresh_in X ϕ
  → svar_is_fresh_in Y ϕ
    → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
      eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
        (* base case - svar *)
        move=> X Y S HfrX HfrY.
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ϕ
HfrY: svar_is_fresh_in Y ϕ
eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
        destruct ϕ; simpl in Hsz; try lia.
signature: Signature
M: Model
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
eval (update_svar_val X S ρ)
  (patt_free_evar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_evar x)^{svar:dbi↦Y}
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
eval (update_svar_val X S ρ)
  (patt_free_svar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_svar x)^{svar:dbi↦Y}
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
eval (update_svar_val X S ρ)
  (patt_bound_evar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_evar n)^{svar:dbi↦Y}
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
eval (update_svar_val X S ρ)
  (patt_bound_svar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_svar n)^{svar:dbi↦Y}
signature: Signature
M: Model
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
eval (update_svar_val X S ρ)
  (patt_sym sigma)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_sym sigma)^{svar:dbi↦Y}
signature: Signature
M: Model
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
eval (update_svar_val X S ρ) ⊥^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) ⊥^{svar:dbi↦Y}
        +
signature: Signature
M: Model
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
eval (update_svar_val X S ρ)
  (patt_free_evar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_evar x)^{svar:dbi↦Y} rewrite 2!eval_free_evar_simpl.
signature: Signature
M: Model
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
{[evar_valuation (update_svar_val X S ρ) x]} =
{[evar_valuation (update_svar_val Y S ρ) x]} reflexivity.
        +
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
eval (update_svar_val X S ρ)
  (patt_free_svar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_svar x)^{svar:dbi↦Y} rewrite 2!eval_free_svar_simpl.
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
svar_valuation (update_svar_val X S ρ) x =
svar_valuation (update_svar_val Y S ρ) x
          unfold svar_is_fresh_in in HfrX, HfrY.
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: X ∉ free_svars (patt_free_svar x)
HfrY: Y ∉ free_svars (patt_free_svar x)
svar_valuation (update_svar_val X S ρ) x =
svar_valuation (update_svar_val Y S ρ) x simpl in HfrX, HfrY.
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: X ∉ {[x]}
HfrY: Y ∉ {[x]}
svar_valuation (update_svar_val X S ρ) x =
svar_valuation (update_svar_val Y S ρ) x
          apply not_elem_of_singleton_1 in HfrX.
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: X ≠ x
HfrY: Y ∉ {[x]}
svar_valuation (update_svar_val X S ρ) x =
svar_valuation (update_svar_val Y S ρ) x
          apply not_elem_of_singleton_1 in HfrY.
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: X ≠ x
HfrY: Y ≠ x
svar_valuation (update_svar_val X S ρ) x =
svar_valuation (update_svar_val Y S ρ) x
          unfold update_svar_val.
signature: Signature
M: Model
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: X ≠ x
HfrY: Y ≠ x
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then S
        else svar_valuation ρ sv'
  |} x =
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (Y = sv')
        then S
        else svar_valuation ρ sv'
  |} x
          destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
x: svar
dbi: db_index
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: X ≠ x
HfrY: Y ≠ x
svar_valuation
  {|
    evar_valuation :=
      evar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |};
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then S
        else
         svar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} sv'
  |} x =
svar_valuation
  {|
    evar_valuation :=
      evar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |};
    svar_valuation :=
      λ sv' : svar,
        if decide (Y = sv')
        then S
        else
         svar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} sv'
  |} x simpl.
signature: Signature
M: Model
x: svar
dbi: db_index
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: X ≠ x
HfrY: Y ≠ x
(if decide (X = x) then S else ρₛ x) =
(if decide (Y = x) then S else ρₛ x)
          destruct (decide (X = x)),(decide (Y = x)); simpl; congruence.
        +
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
eval (update_svar_val X S ρ)
  (patt_bound_evar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_evar n)^{svar:dbi↦Y} rewrite 2!eval_bound_evar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
∅ = ∅ reflexivity.
        +
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
eval (update_svar_val X S ρ)
  (patt_bound_svar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_svar n)^{svar:dbi↦Y} unfold svar_open.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
eval (update_svar_val X S ρ)
  (patt_bound_svar n)^[svar:dbi↦patt_free_svar X] =
eval (update_svar_val Y S ρ)
  (patt_bound_svar n)^[svar:dbi↦patt_free_svar Y] simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
eval (update_svar_val X S ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val Y S ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
eval (update_svar_val X S ρ) (patt_bound_svar n) =
eval (update_svar_val Y S ρ) (patt_bound_svar n)
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
eval (update_svar_val X S ρ) (patt_free_svar X) =
eval (update_svar_val Y S ρ) (patt_free_svar Y)
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
eval (update_svar_val X S ρ)
  (patt_bound_svar (Nat.pred n)) =
eval (update_svar_val Y S ρ)
  (patt_bound_svar (Nat.pred n))
          *
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
eval (update_svar_val X S ρ) (patt_bound_svar n) =
eval (update_svar_val Y S ρ) (patt_bound_svar n) rewrite 2!eval_bound_svar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
∅ = ∅
            reflexivity.
          *
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
eval (update_svar_val X S ρ) (patt_free_svar X) =
eval (update_svar_val Y S ρ) (patt_free_svar Y) rewrite 2!eval_free_svar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
svar_valuation (update_svar_val X S ρ) X =
svar_valuation (update_svar_val Y S ρ) Y
            rewrite 2!update_svar_val_same.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
S = S reflexivity.
          *
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
eval (update_svar_val X S ρ)
  (patt_bound_svar (Nat.pred n)) =
eval (update_svar_val Y S ρ)
  (patt_bound_svar (Nat.pred n)) rewrite 2!eval_bound_svar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
∅ = ∅
            reflexivity.
        +
signature: Signature
M: Model
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
eval (update_svar_val X S ρ)
  (patt_sym sigma)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_sym sigma)^{svar:dbi↦Y} rewrite 2!eval_sym_simpl.
signature: Signature
M: Model
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
sym_interp M sigma = sym_interp M sigma
          reflexivity.
        +
signature: Signature
M: Model
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
eval (update_svar_val X S ρ) ⊥^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) ⊥^{svar:dbi↦Y} rewrite 2!eval_bott_simpl.
signature: Signature
M: Model
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
∅ = ∅
          reflexivity.
      }
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
∀ (x y : evar) (c : M),
  evar_is_fresh_in x ϕ
  → evar_is_fresh_in y ϕ
    → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
      eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
      {
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
∀ (x y : evar) (c : M),
  evar_is_fresh_in x ϕ
  → evar_is_fresh_in y ϕ
    → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
      eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
        (* base case - evar *)
        move=> x y c Hfrx Hfry.
signature: Signature
M: Model
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ϕ
Hfry: evar_is_fresh_in y ϕ
eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
        destruct ϕ; simpl in Hsz; try lia.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
eval (update_evar_val x c ρ)
  (patt_free_evar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_evar x0)^{evar:dbi↦y}
signature: Signature
M: Model
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
eval (update_evar_val x c ρ)
  (patt_free_svar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_svar x0)^{evar:dbi↦y}
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
eval (update_evar_val x c ρ)
  (patt_bound_evar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_evar n)^{evar:dbi↦y}
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
eval (update_evar_val x c ρ)
  (patt_bound_svar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_svar n)^{evar:dbi↦y}
signature: Signature
M: Model
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
eval (update_evar_val x c ρ)
  (patt_sym sigma)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_sym sigma)^{evar:dbi↦y}
signature: Signature
M: Model
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
eval (update_evar_val x c ρ) ⊥^{evar:dbi↦x} =
eval (update_evar_val y c ρ) ⊥^{evar:dbi↦y}
        +
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
eval (update_evar_val x c ρ)
  (patt_free_evar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_evar x0)^{evar:dbi↦y} rewrite 2!eval_free_evar_simpl.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
{[evar_valuation (update_evar_val x c ρ) x0]} =
{[evar_valuation (update_evar_val y c ρ) x0]}
          unfold evar_is_fresh_in in Hfrx, Hfry.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ∉ free_evars (patt_free_evar x0)
Hfry: y ∉ free_evars (patt_free_evar x0)
{[evar_valuation (update_evar_val x c ρ) x0]} =
{[evar_valuation (update_evar_val y c ρ) x0]} simpl in Hfrx, Hfry.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ∉ {[x0]}
Hfry: y ∉ {[x0]}
{[evar_valuation (update_evar_val x c ρ) x0]} =
{[evar_valuation (update_evar_val y c ρ) x0]}
          apply not_elem_of_singleton_1 in Hfrx.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ≠ x0
Hfry: y ∉ {[x0]}
{[evar_valuation (update_evar_val x c ρ) x0]} =
{[evar_valuation (update_evar_val y c ρ) x0]}
          apply not_elem_of_singleton_1 in Hfry.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ≠ x0
Hfry: y ≠ x0
{[evar_valuation (update_evar_val x c ρ) x0]} =
{[evar_valuation (update_evar_val y c ρ) x0]}
          apply f_equal.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ≠ x0
Hfry: y ≠ x0
evar_valuation (update_evar_val x c ρ) x0 =
evar_valuation (update_evar_val y c ρ) x0 unfold update_evar_val.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ≠ x0
Hfry: y ≠ x0
evar_valuation
  {|
    evar_valuation :=
      λ ev' : evar,
        if decide (x = ev')
        then c
        else evar_valuation ρ ev';
    svar_valuation := svar_valuation ρ
  |} x0 =
evar_valuation
  {|
    evar_valuation :=
      λ ev' : evar,
        if decide (y = ev')
        then c
        else evar_valuation ρ ev';
    svar_valuation := svar_valuation ρ
  |} x0
          destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
x0: evar
dbi: db_index
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ≠ x0
Hfry: y ≠ x0
evar_valuation
  {|
    evar_valuation :=
      λ ev' : evar,
        if decide (x = ev')
        then c
        else
         evar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} ev';
    svar_valuation :=
      svar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |}
  |} x0 =
evar_valuation
  {|
    evar_valuation :=
      λ ev' : evar,
        if decide (y = ev')
        then c
        else
         evar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} ev';
    svar_valuation :=
      svar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |}
  |} x0 simpl.
signature: Signature
M: Model
x0: evar
dbi: db_index
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: x ≠ x0
Hfry: y ≠ x0
(if decide (x = x0) then c else ρₑ x0) =
(if decide (y = x0) then c else ρₑ x0)
          destruct (decide (x = x0)),(decide (y = x0)); simpl; congruence.
        +
signature: Signature
M: Model
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
eval (update_evar_val x c ρ)
  (patt_free_svar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_svar x0)^{evar:dbi↦y} rewrite 2!eval_free_svar_simpl.
signature: Signature
M: Model
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
svar_valuation (update_evar_val x c ρ) x0 =
svar_valuation (update_evar_val y c ρ) x0
          reflexivity.
        +
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
eval (update_evar_val x c ρ)
  (patt_bound_evar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_evar n)^{evar:dbi↦y} unfold evar_open.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
eval (update_evar_val x c ρ)
  (patt_bound_evar n)^[evar:dbi↦patt_free_evar x] =
eval (update_evar_val y c ρ)
  (patt_bound_evar n)^[evar:dbi↦patt_free_evar y] simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
eval (update_evar_val x c ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end =
eval (update_evar_val y c ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar y
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end case_match.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
eval (update_evar_val x c ρ) (patt_bound_evar n) =
eval (update_evar_val y c ρ) (patt_bound_evar n)
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
eval (update_evar_val x c ρ) (patt_free_evar x) =
eval (update_evar_val y c ρ) (patt_free_evar y)
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
eval (update_evar_val x c ρ)
  (patt_bound_evar (Nat.pred n)) =
eval (update_evar_val y c ρ)
  (patt_bound_evar (Nat.pred n))
          *
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
eval (update_evar_val x c ρ) (patt_bound_evar n) =
eval (update_evar_val y c ρ) (patt_bound_evar n) rewrite 2!eval_bound_evar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
∅ = ∅
            reflexivity.
          *
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
eval (update_evar_val x c ρ) (patt_free_evar x) =
eval (update_evar_val y c ρ) (patt_free_evar y) rewrite 2!eval_free_evar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
{[evar_valuation (update_evar_val x c ρ) x]} =
{[evar_valuation (update_evar_val y c ρ) y]}
            rewrite 2!update_evar_val_same.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
{[c]} = {[c]} reflexivity.
          *
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
eval (update_evar_val x c ρ)
  (patt_bound_evar (Nat.pred n)) =
eval (update_evar_val y c ρ)
  (patt_bound_evar (Nat.pred n)) rewrite 2!eval_bound_evar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
∅ = ∅
            reflexivity.
        +
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
eval (update_evar_val x c ρ)
  (patt_bound_svar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_svar n)^{evar:dbi↦y} rewrite 2!eval_bound_svar_simpl.
signature: Signature
M: Model
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
∅ = ∅ reflexivity.
        +
signature: Signature
M: Model
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
eval (update_evar_val x c ρ)
  (patt_sym sigma)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_sym sigma)^{evar:dbi↦y} rewrite 2!eval_sym_simpl.
signature: Signature
M: Model
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
sym_interp M sigma = sym_interp M sigma
          reflexivity.
        +
signature: Signature
M: Model
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
eval (update_evar_val x c ρ) ⊥^{evar:dbi↦x} =
eval (update_evar_val y c ρ) ⊥^{evar:dbi↦y} rewrite 2!eval_bott_simpl.
signature: Signature
M: Model
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ 0
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
∅ = ∅
          reflexivity.
      }
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
∀ (ϕ : Pattern) (dbi : db_index) (ρ : Valuation),
  size ϕ ≤ S sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y}) move=> ϕ dbi ρ Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ S sz
(∀ (X Y : svar) (S : propset M),
   svar_is_fresh_in X ϕ
   → svar_is_fresh_in Y ϕ
     → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
       eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y})
∧ (∀ (x y : evar) (c : M),
     evar_is_fresh_in x ϕ
     → evar_is_fresh_in y ϕ
       → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
         eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y})
      split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ S sz
∀ (X Y : svar) (S : propset M),
  svar_is_fresh_in X ϕ
  → svar_is_fresh_in Y ϕ
    → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
      eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ S sz
∀ (x y : evar) (c : M),
  evar_is_fresh_in x ϕ
  → evar_is_fresh_in y ϕ
    → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
      eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
      {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ S sz
∀ (X Y : svar) (S : propset M),
  svar_is_fresh_in X ϕ
  → svar_is_fresh_in Y ϕ
    → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
      eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
        (* inductive case - svar *)
        move=> X Y S HfrX HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ϕ
HfrY: svar_is_fresh_in Y ϕ
eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
        destruct ϕ; simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
eval (update_svar_val X S ρ)
  (patt_free_evar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_evar x)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
eval (update_svar_val X S ρ)
  (patt_free_svar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_svar x)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
eval (update_svar_val X S ρ)
  (patt_bound_evar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_evar n)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
eval (update_svar_val X S ρ)
  (patt_bound_svar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_svar n)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
eval (update_svar_val X S ρ)
  (patt_sym sigma)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_sym sigma)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
eval (update_svar_val X S ρ) (ϕ1 $ ϕ2)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (ϕ1 $ ϕ2)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
eval (update_svar_val X S ρ) ⊥^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) ⊥^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
eval (update_svar_val X S ρ) (ϕ1 ---> ϕ2)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (ϕ1 ---> ϕ2)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
eval (update_svar_val X S ρ) (ex , ϕ)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (ex , ϕ)^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
eval (update_svar_val X S ρ) (mu , ϕ)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (mu , ϕ)^{svar:dbi↦Y}
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
eval (update_svar_val X S ρ)
  (patt_free_evar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_evar x)^{svar:dbi↦Y} rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
size (patt_free_evar x) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
svar_is_fresh_in X (patt_free_evar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
svar_is_fresh_in Y (patt_free_evar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
eval (update_svar_val Y S ρ)
  (patt_free_evar x)^{svar:dbi↦Y} =
eval (update_svar_val Y S ρ)
  (patt_free_evar x)^{svar:dbi↦Y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
size (patt_free_evar x) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
svar_is_fresh_in X (patt_free_evar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
svar_is_fresh_in Y (patt_free_evar x) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
size (patt_free_evar x) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_evar x)
HfrY: svar_is_fresh_in Y (patt_free_evar x)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
eval (update_svar_val X S ρ)
  (patt_free_svar x)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_free_svar x)^{svar:dbi↦Y} rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
size (patt_free_svar x) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
svar_is_fresh_in X (patt_free_svar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
svar_is_fresh_in Y (patt_free_svar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
eval (update_svar_val Y S ρ)
  (patt_free_svar x)^{svar:dbi↦Y} =
eval (update_svar_val Y S ρ)
  (patt_free_svar x)^{svar:dbi↦Y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
size (patt_free_svar x) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
svar_is_fresh_in X (patt_free_svar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
svar_is_fresh_in Y (patt_free_svar x) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
size (patt_free_svar x) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_free_svar x)
HfrY: svar_is_fresh_in Y (patt_free_svar x)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
eval (update_svar_val X S ρ)
  (patt_bound_evar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_evar n)^{svar:dbi↦Y} rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
size (patt_bound_evar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
svar_is_fresh_in X (patt_bound_evar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
svar_is_fresh_in Y (patt_bound_evar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
eval (update_svar_val Y S ρ)
  (patt_bound_evar n)^{svar:dbi↦Y} =
eval (update_svar_val Y S ρ)
  (patt_bound_evar n)^{svar:dbi↦Y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
size (patt_bound_evar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
svar_is_fresh_in X (patt_bound_evar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
svar_is_fresh_in Y (patt_bound_evar n) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
size (patt_bound_evar n) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_evar n)
HfrY: svar_is_fresh_in Y (patt_bound_evar n)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
eval (update_svar_val X S ρ)
  (patt_bound_svar n)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_bound_svar n)^{svar:dbi↦Y} rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
size (patt_bound_svar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
svar_is_fresh_in X (patt_bound_svar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
svar_is_fresh_in Y (patt_bound_svar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
eval (update_svar_val Y S ρ)
  (patt_bound_svar n)^{svar:dbi↦Y} =
eval (update_svar_val Y S ρ)
  (patt_bound_svar n)^{svar:dbi↦Y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
size (patt_bound_svar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
svar_is_fresh_in X (patt_bound_svar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
svar_is_fresh_in Y (patt_bound_svar n) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
size (patt_bound_svar n) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_bound_svar n)
HfrY: svar_is_fresh_in Y (patt_bound_svar n)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
eval (update_svar_val X S ρ)
  (patt_sym sigma)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ)
  (patt_sym sigma)^{svar:dbi↦Y} rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
size (patt_sym sigma) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
svar_is_fresh_in X (patt_sym sigma)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
svar_is_fresh_in Y (patt_sym sigma)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
eval (update_svar_val Y S ρ)
  (patt_sym sigma)^{svar:dbi↦Y} =
eval (update_svar_val Y S ρ)
  (patt_sym sigma)^{svar:dbi↦Y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
size (patt_sym sigma) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
svar_is_fresh_in X (patt_sym sigma)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
svar_is_fresh_in Y (patt_sym sigma) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
size (patt_sym sigma) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (patt_sym sigma)
HfrY: svar_is_fresh_in Y (patt_sym sigma)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
eval (update_svar_val X S ρ) (ϕ1 $ ϕ2)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (ϕ1 $ ϕ2)^{svar:dbi↦Y} rewrite 2!eval_app_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext
  (eval (update_svar_val X S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ1))
  (eval (update_svar_val X S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ2)) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2)) fold svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext
  (eval (update_svar_val X S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ1))
  (eval (update_svar_val X S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ2)) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2))
          rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val X S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ2)) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2)) 4: rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2)) lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2))
          eapply svar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2)) apply HfrX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2))
          eapply svar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2)) apply HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2))
          lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2))
          eapply svar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in X (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2)) apply HfrX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2))
          eapply svar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
svar_is_fresh_in Y (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2)) apply HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 $ ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 $ ϕ2)
app_ext (eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y})
  (eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y}) =
app_ext
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ1))
  (eval (update_svar_val Y S ρ)
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ2))
          reflexivity.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
eval (update_svar_val X S ρ) ⊥^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) ⊥^{svar:dbi↦Y} rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
size ⊥ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
svar_is_fresh_in X ⊥
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
svar_is_fresh_in Y ⊥
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
eval (update_svar_val Y S ρ) ⊥^{svar:dbi↦Y} =
eval (update_svar_val Y S ρ) ⊥^{svar:dbi↦Y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
size ⊥ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
svar_is_fresh_in X ⊥
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
svar_is_fresh_in Y ⊥ 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
size ⊥ ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X ⊥
HfrY: svar_is_fresh_in Y ⊥
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
eval (update_svar_val X S ρ) (ϕ1 ---> ϕ2)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (ϕ1 ---> ϕ2)^{svar:dbi↦Y} rewrite 2!eval_imp_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤
∖ eval (update_svar_val X S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar X) dbi ϕ1)
∪ eval (update_svar_val X S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar X) dbi ϕ2) =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2) fold svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤
∖ eval (update_svar_val X S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar X) dbi ϕ1)
∪ eval (update_svar_val X S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar X) dbi ϕ2) =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2)
          rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val X S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar X) dbi ϕ2) =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2) 4: rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2) lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2)
          eapply svar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2) apply HfrX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2)
          eapply svar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2) apply HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2)
          lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2)
          eapply svar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in X (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2) apply HfrX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2)
          eapply svar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
svar_is_fresh_in Y (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2) apply HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ1 + size ϕ2) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ϕ1 ---> ϕ2)
HfrY: svar_is_fresh_in Y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_svar_val Y S ρ) ϕ1^{svar:dbi↦Y}
∪ eval (update_svar_val Y S ρ) ϕ2^{svar:dbi↦Y} =
⊤
∖ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ1)
∪ eval (update_svar_val Y S ρ)
    ((fix bsvar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x phi1 $
            bsvar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x phi1 --->
            bsvar_subst psi x phi2
        | ex , phi' => ex , bsvar_subst psi x phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x) phi'
        end) (patt_free_svar Y) dbi ϕ2)
          reflexivity.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
eval (update_svar_val X S ρ) (ex , ϕ)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (ex , ϕ)^{svar:dbi↦Y} rewrite 2!eval_ex_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
(let x :=
   fresh_evar
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ) in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val X S ρ) in
      eval ρ'
        ((fix bsvar_subst
            (psi : Pattern) (x0 : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x0 with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x0 phi1 $
                bsvar_subst psi x0 phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x0 phi1 --->
                bsvar_subst psi x0 phi2
            | ex , phi' =>
                ex , bsvar_subst psi x0 phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x0) phi'
            end) (patt_free_svar X) dbi ϕ)^{evar:0↦x})) =
(let x :=
   fresh_evar
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ) in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val Y S ρ) in
      eval ρ'
        ((fix bsvar_subst
            (psi : Pattern) (x0 : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x0 with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x0 phi1 $
                bsvar_subst psi x0 phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x0 phi1 --->
                bsvar_subst psi x0 phi2
            | ex , phi' =>
                ex , bsvar_subst psi x0 phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x0) phi'
            end) (patt_free_svar Y) dbi ϕ)^{evar:0↦x})) fold svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
(let x :=
   fresh_evar
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ) in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val X S ρ) in
      eval ρ'
        ((fix bsvar_subst
            (psi : Pattern) (x0 : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x0 with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x0 phi1 $
                bsvar_subst psi x0 phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x0 phi1 --->
                bsvar_subst psi x0 phi2
            | ex , phi' =>
                ex , bsvar_subst psi x0 phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x0) phi'
            end) (patt_free_svar X) dbi ϕ)^{evar:0↦x})) =
(let x :=
   fresh_evar
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ) in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_svar_val Y S ρ) in
      eval ρ'
        ((fix bsvar_subst
            (psi : Pattern) (x0 : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x0 with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x0 phi1 $
                bsvar_subst psi x0 phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x0 phi1 --->
                bsvar_subst psi x0 phi2
            | ex , phi' =>
                ex , bsvar_subst psi x0 phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x0) phi'
            end) (patt_free_svar Y) dbi ϕ)^{evar:0↦x})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar
             ((fix bsvar_subst
                 (psi : Pattern) (x : db_index)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x' =>
                     patt_free_evar x'
                 | patt_free_svar x' =>
                     patt_free_svar x'
                 | patt_bound_evar n =>
                     patt_bound_evar n
                 | patt_bound_svar n =>
                     match compare_nat n x with
                     | Nat_less _ _ _ =>
                         patt_bound_svar n
                     | Nat_equal _ _ _ => psi
                     | Nat_greater _ _ _ =>
                         patt_bound_svar (Nat.pred n)
                     end
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     bsvar_subst psi x phi1 $
                     bsvar_subst psi x phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     bsvar_subst psi x phi1 --->
                     bsvar_subst psi x phi2
                 | ex , phi' =>
                     ex , bsvar_subst psi x phi'
                 | mu , phi' =>
                     mu ,
                     bsvar_subst psi (Datatypes.S x)
                       phi'
                 end) (patt_free_svar X) dbi ϕ)) e
          (update_svar_val X S ρ))
       ((fix bsvar_subst
           (psi : Pattern) (x : db_index)
           (phi : Pattern) {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x' => patt_free_evar x'
           | patt_free_svar x' => patt_free_svar x'
           | patt_bound_evar n => patt_bound_evar n
           | patt_bound_svar n =>
               match compare_nat n x with
               | Nat_less _ _ _ => patt_bound_svar n
               | Nat_equal _ _ _ => psi
               | Nat_greater _ _ _ =>
                   patt_bound_svar (Nat.pred n)
               end
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               bsvar_subst psi x phi1 $
               bsvar_subst psi x phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               bsvar_subst psi x phi1 --->
               bsvar_subst psi x phi2
           | ex , phi' => ex , bsvar_subst psi x phi'
           | mu , phi' =>
               mu ,
               bsvar_subst psi (Datatypes.S x) phi'
           end) (patt_free_svar X) dbi ϕ)^{evar:0↦
       fresh_evar
         ((fix bsvar_subst
             (psi : Pattern) (x : db_index)
             (phi : Pattern) {struct phi} : Pattern :=
             match phi with
             | patt_free_evar x' => patt_free_evar x'
             | patt_free_svar x' => patt_free_svar x'
             | patt_bound_evar n => patt_bound_evar n
             | patt_bound_svar n =>
                 match compare_nat n x with
                 | Nat_less _ _ _ => patt_bound_svar n
                 | Nat_equal _ _ _ => psi
                 | Nat_greater _ _ _ =>
                     patt_bound_svar (Nat.pred n)
                 end
             | patt_sym sigma => patt_sym sigma
             | phi1 $ phi2 =>
                 bsvar_subst psi x phi1 $
                 bsvar_subst psi x phi2
             | ⊥ => ⊥
             | phi1 ---> phi2 =>
                 bsvar_subst psi x phi1 --->
                 bsvar_subst psi x phi2
             | ex , phi' =>
                 ex , bsvar_subst psi x phi'
             | mu , phi' =>
                 mu ,
                 bsvar_subst psi (Datatypes.S x) phi'
             end) (patt_free_svar X) dbi ϕ)}) =
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar
             ((fix bsvar_subst
                 (psi : Pattern) (x : db_index)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x' =>
                     patt_free_evar x'
                 | patt_free_svar x' =>
                     patt_free_svar x'
                 | patt_bound_evar n =>
                     patt_bound_evar n
                 | patt_bound_svar n =>
                     match compare_nat n x with
                     | Nat_less _ _ _ =>
                         patt_bound_svar n
                     | Nat_equal _ _ _ => psi
                     | Nat_greater _ _ _ =>
                         patt_bound_svar (Nat.pred n)
                     end
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     bsvar_subst psi x phi1 $
                     bsvar_subst psi x phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     bsvar_subst psi x phi1 --->
                     bsvar_subst psi x phi2
                 | ex , phi' =>
                     ex , bsvar_subst psi x phi'
                 | mu , phi' =>
                     mu ,
                     bsvar_subst psi (Datatypes.S x)
                       phi'
                 end) (patt_free_svar Y) dbi ϕ)) e
          (update_svar_val Y S ρ))
       ((fix bsvar_subst
           (psi : Pattern) (x : db_index)
           (phi : Pattern) {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x' => patt_free_evar x'
           | patt_free_svar x' => patt_free_svar x'
           | patt_bound_evar n => patt_bound_evar n
           | patt_bound_svar n =>
               match compare_nat n x with
               | Nat_less _ _ _ => patt_bound_svar n
               | Nat_equal _ _ _ => psi
               | Nat_greater _ _ _ =>
                   patt_bound_svar (Nat.pred n)
               end
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               bsvar_subst psi x phi1 $
               bsvar_subst psi x phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               bsvar_subst psi x phi1 --->
               bsvar_subst psi x phi2
           | ex , phi' => ex , bsvar_subst psi x phi'
           | mu , phi' =>
               mu ,
               bsvar_subst psi (Datatypes.S x) phi'
           end) (patt_free_svar Y) dbi ϕ)^{evar:0↦
       fresh_evar
         ((fix bsvar_subst
             (psi : Pattern) (x : db_index)
             (phi : Pattern) {struct phi} : Pattern :=
             match phi with
             | patt_free_evar x' => patt_free_evar x'
             | patt_free_svar x' => patt_free_svar x'
             | patt_bound_evar n => patt_bound_evar n
             | patt_bound_svar n =>
                 match compare_nat n x with
                 | Nat_less _ _ _ => patt_bound_svar n
                 | Nat_equal _ _ _ => psi
                 | Nat_greater _ _ _ =>
                     patt_bound_svar (Nat.pred n)
                 end
             | patt_sym sigma => patt_sym sigma
             | phi1 $ phi2 =>
                 bsvar_subst psi x phi1 $
                 bsvar_subst psi x phi2
             | ⊥ => ⊥
             | phi1 ---> phi2 =>
                 bsvar_subst psi x phi1 --->
                 bsvar_subst psi x phi2
             | ex , phi' =>
                 ex , bsvar_subst psi x phi'
             | mu , phi' =>
                 mu ,
                 bsvar_subst psi (Datatypes.S x) phi'
             end) (patt_free_svar Y) dbi ϕ)})
          apply f_equal.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
(λ e : M,
   eval
     (update_evar_val
        (fresh_evar
           ((fix bsvar_subst
               (psi : Pattern) (x : db_index)
               (phi : Pattern) {struct phi} :
                 Pattern :=
               match phi with
               | patt_free_evar x' =>
                   patt_free_evar x'
               | patt_free_svar x' =>
                   patt_free_svar x'
               | patt_bound_evar n =>
                   patt_bound_evar n
               | patt_bound_svar n =>
                   match compare_nat n x with
                   | Nat_less _ _ _ =>
                       patt_bound_svar n
                   | Nat_equal _ _ _ => psi
                   | Nat_greater _ _ _ =>
                       patt_bound_svar (Nat.pred n)
                   end
               | patt_sym sigma => patt_sym sigma
               | phi1 $ phi2 =>
                   bsvar_subst psi x phi1 $
                   bsvar_subst psi x phi2
               | ⊥ => ⊥
               | phi1 ---> phi2 =>
                   bsvar_subst psi x phi1 --->
                   bsvar_subst psi x phi2
               | ex , phi' =>
                   ex , bsvar_subst psi x phi'
               | mu , phi' =>
                   mu ,
                   bsvar_subst psi (Datatypes.S x)
                     phi'
               end) (patt_free_svar X) dbi ϕ)) e
        (update_svar_val X S ρ))
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar X) dbi ϕ)^{evar:0↦fresh_evar
                                                 ((fix
                                                 bsvar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 patt_bound_evar
                                                 n
                                                 | patt_bound_svar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 $
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 --->
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bsvar_subst
                                                 psi x
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bsvar_subst
                                                 psi
                                                 (Datatypes.S
                                                 x)
                                                 phi'
                                                 end)
                                                 (patt_free_svar
                                                 X)
                                                 dbi ϕ)}) =
(λ e : M,
   eval
     (update_evar_val
        (fresh_evar
           ((fix bsvar_subst
               (psi : Pattern) (x : db_index)
               (phi : Pattern) {struct phi} :
                 Pattern :=
               match phi with
               | patt_free_evar x' =>
                   patt_free_evar x'
               | patt_free_svar x' =>
                   patt_free_svar x'
               | patt_bound_evar n =>
                   patt_bound_evar n
               | patt_bound_svar n =>
                   match compare_nat n x with
                   | Nat_less _ _ _ =>
                       patt_bound_svar n
                   | Nat_equal _ _ _ => psi
                   | Nat_greater _ _ _ =>
                       patt_bound_svar (Nat.pred n)
                   end
               | patt_sym sigma => patt_sym sigma
               | phi1 $ phi2 =>
                   bsvar_subst psi x phi1 $
                   bsvar_subst psi x phi2
               | ⊥ => ⊥
               | phi1 ---> phi2 =>
                   bsvar_subst psi x phi1 --->
                   bsvar_subst psi x phi2
               | ex , phi' =>
                   ex , bsvar_subst psi x phi'
               | mu , phi' =>
                   mu ,
                   bsvar_subst psi (Datatypes.S x)
                     phi'
               end) (patt_free_svar Y) dbi ϕ)) e
        (update_svar_val Y S ρ))
     ((fix bsvar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n => patt_bound_evar n
         | patt_bound_svar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_svar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_svar (Nat.pred n)
             end
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bsvar_subst psi x phi1 $
             bsvar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bsvar_subst psi x phi1 --->
             bsvar_subst psi x phi2
         | ex , phi' => ex , bsvar_subst psi x phi'
         | mu , phi' =>
             mu , bsvar_subst psi (Datatypes.S x) phi'
         end) (patt_free_svar Y) dbi ϕ)^{evar:0↦fresh_evar
                                                 ((fix
                                                 bsvar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 patt_bound_evar
                                                 n
                                                 | patt_bound_svar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 $
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 --->
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bsvar_subst
                                                 psi x
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bsvar_subst
                                                 psi
                                                 (Datatypes.S
                                                 x)
                                                 phi'
                                                 end)
                                                 (patt_free_svar
                                                 Y)
                                                 dbi ϕ)}) apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
∀ x : M,
  eval
    (update_evar_val
       (fresh_evar
          ((fix bsvar_subst
              (psi : Pattern) (x0 : db_index)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x' => patt_free_evar x'
              | patt_free_svar x' => patt_free_svar x'
              | patt_bound_evar n => patt_bound_evar n
              | patt_bound_svar n =>
                  match compare_nat n x0 with
                  | Nat_less _ _ _ =>
                      patt_bound_svar n
                  | Nat_equal _ _ _ => psi
                  | Nat_greater _ _ _ =>
                      patt_bound_svar (Nat.pred n)
                  end
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  bsvar_subst psi x0 phi1 $
                  bsvar_subst psi x0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  bsvar_subst psi x0 phi1 --->
                  bsvar_subst psi x0 phi2
              | ex , phi' =>
                  ex , bsvar_subst psi x0 phi'
              | mu , phi' =>
                  mu ,
                  bsvar_subst psi (Datatypes.S x0)
                    phi'
              end) (patt_free_svar X) dbi ϕ)) x
       (update_svar_val X S ρ))
    ((fix bsvar_subst
        (psi : Pattern) (x0 : db_index)
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x0 with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x0 phi1 $
            bsvar_subst psi x0 phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x0 phi1 --->
            bsvar_subst psi x0 phi2
        | ex , phi' => ex , bsvar_subst psi x0 phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x0) phi'
        end) (patt_free_svar X) dbi ϕ)^{evar:0↦fresh_evar
                                                 ((fix
                                                 bsvar_subst
                                                 (psi : Pattern)
                                                 (x0 : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 patt_bound_evar
                                                 n
                                                 | patt_bound_svar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x0
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi1 $
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi1 --->
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bsvar_subst
                                                 psi
                                                 (Datatypes.S
                                                 x0)
                                                 phi'
                                                 end)
                                                 (patt_free_svar
                                                 X)
                                                 dbi ϕ)} =
  eval
    (update_evar_val
       (fresh_evar
          ((fix bsvar_subst
              (psi : Pattern) (x0 : db_index)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x' => patt_free_evar x'
              | patt_free_svar x' => patt_free_svar x'
              | patt_bound_evar n => patt_bound_evar n
              | patt_bound_svar n =>
                  match compare_nat n x0 with
                  | Nat_less _ _ _ =>
                      patt_bound_svar n
                  | Nat_equal _ _ _ => psi
                  | Nat_greater _ _ _ =>
                      patt_bound_svar (Nat.pred n)
                  end
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  bsvar_subst psi x0 phi1 $
                  bsvar_subst psi x0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  bsvar_subst psi x0 phi1 --->
                  bsvar_subst psi x0 phi2
              | ex , phi' =>
                  ex , bsvar_subst psi x0 phi'
              | mu , phi' =>
                  mu ,
                  bsvar_subst psi (Datatypes.S x0)
                    phi'
              end) (patt_free_svar Y) dbi ϕ)) x
       (update_svar_val Y S ρ))
    ((fix bsvar_subst
        (psi : Pattern) (x0 : db_index)
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n => patt_bound_evar n
        | patt_bound_svar n =>
            match compare_nat n x0 with
            | Nat_less _ _ _ => patt_bound_svar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_svar (Nat.pred n)
            end
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bsvar_subst psi x0 phi1 $
            bsvar_subst psi x0 phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bsvar_subst psi x0 phi1 --->
            bsvar_subst psi x0 phi2
        | ex , phi' => ex , bsvar_subst psi x0 phi'
        | mu , phi' =>
            mu , bsvar_subst psi (Datatypes.S x0) phi'
        end) (patt_free_svar Y) dbi ϕ)^{evar:0↦fresh_evar
                                                 ((fix
                                                 bsvar_subst
                                                 (psi : Pattern)
                                                 (x0 : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 patt_bound_evar
                                                 n
                                                 | patt_bound_svar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x0
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi1 $
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi1 --->
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bsvar_subst
                                                 psi
                                                 x0
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bsvar_subst
                                                 psi
                                                 (Datatypes.S
                                                 x0)
                                                 phi'
                                                 end)
                                                 (patt_free_svar
                                                 Y)
                                                 dbi ϕ)} intros e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val
     (fresh_evar
        ((fix bsvar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x phi1 $
                bsvar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x phi1 --->
                bsvar_subst psi x phi2
            | ex , phi' => ex , bsvar_subst psi x phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x) phi'
            end) (patt_free_svar X) dbi ϕ)) e
     (update_svar_val X S ρ))
  ((fix bsvar_subst
      (psi : Pattern) (x : db_index) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n => patt_bound_evar n
      | patt_bound_svar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_svar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_svar (Nat.pred n)
          end
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          bsvar_subst psi x phi1 $
          bsvar_subst psi x phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          bsvar_subst psi x phi1 --->
          bsvar_subst psi x phi2
      | ex , phi' => ex , bsvar_subst psi x phi'
      | mu , phi' =>
          mu , bsvar_subst psi (Datatypes.S x) phi'
      end) (patt_free_svar X) dbi ϕ)^{evar:0↦fresh_evar
                                               ((fix
                                                 bsvar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 patt_bound_evar
                                                 n
                                                 | patt_bound_svar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 $
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 --->
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bsvar_subst
                                                 psi x
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bsvar_subst
                                                 psi
                                                 (Datatypes.S
                                                 x)
                                                 phi'
                                                 end)
                                                 (patt_free_svar
                                                 X)
                                                 dbi ϕ)} =
eval
  (update_evar_val
     (fresh_evar
        ((fix bsvar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x phi1 $
                bsvar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x phi1 --->
                bsvar_subst psi x phi2
            | ex , phi' => ex , bsvar_subst psi x phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x) phi'
            end) (patt_free_svar Y) dbi ϕ)) e
     (update_svar_val Y S ρ))
  ((fix bsvar_subst
      (psi : Pattern) (x : db_index) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n => patt_bound_evar n
      | patt_bound_svar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_svar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_svar (Nat.pred n)
          end
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          bsvar_subst psi x phi1 $
          bsvar_subst psi x phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          bsvar_subst psi x phi1 --->
          bsvar_subst psi x phi2
      | ex , phi' => ex , bsvar_subst psi x phi'
      | mu , phi' =>
          mu , bsvar_subst psi (Datatypes.S x) phi'
      end) (patt_free_svar Y) dbi ϕ)^{evar:0↦fresh_evar
                                               ((fix
                                                 bsvar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 patt_bound_evar
                                                 n
                                                 | patt_bound_svar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_svar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 $
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bsvar_subst
                                                 psi x
                                                 phi1 --->
                                                 bsvar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bsvar_subst
                                                 psi x
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bsvar_subst
                                                 psi
                                                 (Datatypes.S
                                                 x)
                                                 phi'
                                                 end)
                                                 (patt_free_svar
                                                 Y)
                                                 dbi ϕ)}
          rewrite 2!svar_open_evar_open_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val
     (fresh_evar
        ((fix bsvar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x phi1 $
                bsvar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x phi1 --->
                bsvar_subst psi x phi2
            | ex , phi' => ex , bsvar_subst psi x phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x) phi'
            end) (patt_free_svar X) dbi ϕ)) e
     (update_svar_val X S ρ))
  ϕ^{evar:0↦fresh_evar
              ((fix bsvar_subst
                  (psi : Pattern) (x : db_index)
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x' =>
                      patt_free_evar x'
                  | patt_free_svar x' =>
                      patt_free_svar x'
                  | patt_bound_evar n =>
                      patt_bound_evar n
                  | patt_bound_svar n =>
                      match compare_nat n x with
                      | Nat_less _ _ _ =>
                          patt_bound_svar n
                      | Nat_equal _ _ _ => psi
                      | Nat_greater _ _ _ =>
                          patt_bound_svar (Nat.pred n)
                      end
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                      bsvar_subst psi x phi1 $
                      bsvar_subst psi x phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                      bsvar_subst psi x phi1 --->
                      bsvar_subst psi x phi2
                  | ex , phi' =>
                      ex , bsvar_subst psi x phi'
                  | mu , phi' =>
                      mu ,
                      bsvar_subst psi (Datatypes.S x)
                        phi'
                  end) (patt_free_svar X) dbi ϕ)}^{svar:dbi↦X} =
eval
  (update_evar_val
     (fresh_evar
        ((fix bsvar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n => patt_bound_evar n
            | patt_bound_svar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_svar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_svar (Nat.pred n)
                end
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bsvar_subst psi x phi1 $
                bsvar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bsvar_subst psi x phi1 --->
                bsvar_subst psi x phi2
            | ex , phi' => ex , bsvar_subst psi x phi'
            | mu , phi' =>
                mu ,
                bsvar_subst psi (Datatypes.S x) phi'
            end) (patt_free_svar Y) dbi ϕ)) e
     (update_svar_val Y S ρ))
  ϕ^{evar:0↦fresh_evar
              ((fix bsvar_subst
                  (psi : Pattern) (x : db_index)
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x' =>
                      patt_free_evar x'
                  | patt_free_svar x' =>
                      patt_free_svar x'
                  | patt_bound_evar n =>
                      patt_bound_evar n
                  | patt_bound_svar n =>
                      match compare_nat n x with
                      | Nat_less _ _ _ =>
                          patt_bound_svar n
                      | Nat_equal _ _ _ => psi
                      | Nat_greater _ _ _ =>
                          patt_bound_svar (Nat.pred n)
                      end
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                      bsvar_subst psi x phi1 $
                      bsvar_subst psi x phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                      bsvar_subst psi x phi1 --->
                      bsvar_subst psi x phi2
                  | ex , phi' =>
                      ex , bsvar_subst psi x phi'
                  | mu , phi' =>
                      mu ,
                      bsvar_subst psi (Datatypes.S x)
                        phi'
                  end) (patt_free_svar Y) dbi ϕ)}^{svar:dbi↦Y}
          fold bsvar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e
     (update_svar_val X S ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}^{svar:dbi↦X} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]}^{svar:dbi↦Y}
          simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e
     (update_svar_val X S ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}^{svar:dbi↦X} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]}^{svar:dbi↦Y} rewrite 2!update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val X S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}^{svar:dbi↦X} =
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]}^{svar:dbi↦Y}
          rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
size
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in X
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in Y
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]}^{svar:dbi↦Y}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
size
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
≤ sz rewrite -evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
size ϕ ≤ sz lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in X
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in Y
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]}^{svar:dbi↦Y}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in X
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]} apply svar_is_fresh_in_exists in HfrX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (ex , ϕ))
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in X
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
            apply svar_is_fresh_in_exists in HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (ex , ϕ))
HfrY: svar_is_fresh_in Y (ex , (ex , ϕ))
e: M
svar_is_fresh_in X
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
            apply svar_fresh_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (ex , ϕ))
HfrY: svar_is_fresh_in Y (ex , (ex , ϕ))
e: M
svar_is_fresh_in X ϕ apply HfrX.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in Y
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]}^{svar:dbi↦Y}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in Y
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]} apply svar_fresh_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
svar_is_fresh_in Y ϕ apply HfrY. 
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar X]}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^{evar:0↦fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]}^{svar:dbi↦Y}
          rewrite -2!svar_open_evar_open_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar
                           ϕ^[svar:dbi↦patt_free_svar
                                         X]} =
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar
                           ϕ^[svar:dbi↦patt_free_svar
                                         Y]}
          rewrite -2!update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar X]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar
                           ϕ^[svar:dbi↦patt_free_svar
                                         X]} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar
                           ϕ^[svar:dbi↦patt_free_svar
                                         Y]}
          rewrite (proj2 (IHsz _ _ _ _) _ (fresh_evar (ϕ^{svar: dbi ↦ Y}))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
size ϕ^{svar:dbi↦Y} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in
  (fresh_evar ϕ^[svar:dbi↦patt_free_svar X])
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ^{svar:dbi↦Y})
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^[svar:dbi↦
    patt_free_svar Y]} rewrite -svar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in
  (fresh_evar ϕ^[svar:dbi↦patt_free_svar X])
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ^{svar:dbi↦Y})
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^[svar:dbi↦
    patt_free_svar Y]} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in
  (fresh_evar ϕ^[svar:dbi↦patt_free_svar X])
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ^{svar:dbi↦Y})
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^[svar:dbi↦
    patt_free_svar Y]}
          rewrite fresh_evar_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ) ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ^{svar:dbi↦Y})
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^[svar:dbi↦
    patt_free_svar Y]}
          apply evar_fresh_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ) ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ^{svar:dbi↦Y})
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^[svar:dbi↦
    patt_free_svar Y]} apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
evar_is_fresh_in (fresh_evar ϕ^{svar:dbi↦Y})
  ϕ^{svar:dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦
  fresh_evar ϕ^[svar:dbi↦
    patt_free_svar Y]}
          apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_evar_val
     (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e
     (update_svar_val Y S ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar
                           ϕ^[svar:dbi↦patt_free_svar
                                         Y]}
          repeat rewrite update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_svar_val Y S
     (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar
                           ϕ^[svar:dbi↦patt_free_svar
                                         Y]}
          fold bsvar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar ϕ^{svar:dbi↦Y}} =
eval
  (update_svar_val Y S
     (update_evar_val (fresh_evar ϕ^{svar:dbi↦Y}) e ρ))
  ϕ^{svar:dbi↦Y}^{evar:0↦fresh_evar
                           ϕ^[svar:dbi↦patt_free_svar
                                         Y]}
          unfold svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , ϕ)
HfrY: svar_is_fresh_in Y (ex , ϕ)
e: M
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^[svar:dbi↦patt_free_svar Y]^{evar:0↦fresh_evar
                                          ϕ^[svar:dbi↦
                                          patt_free_svar
                                            Y]} =
eval
  (update_svar_val Y S
     (update_evar_val
        (fresh_evar ϕ^[svar:dbi↦patt_free_svar Y]) e ρ))
  ϕ^[svar:dbi↦patt_free_svar Y]^{evar:0↦fresh_evar
                                          ϕ^[svar:dbi↦
                                          patt_free_svar
                                            Y]}
          reflexivity.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
eval (update_svar_val X S ρ) (mu , ϕ)^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) (mu , ϕ)^{svar:dbi↦Y} mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
eval (update_svar_val X S ρ)
  (mu , ϕ^{svar:increase_mu pm_spec_data dbi↦X}) =
eval (update_svar_val Y S ρ)
  (mu , ϕ^{svar:increase_mu pm_spec_data dbi↦Y}) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
eval (update_svar_val X S ρ)
  (mu , ϕ^{svar:Datatypes.S dbi↦X}) =
eval (update_svar_val Y S ρ)
  (mu , ϕ^{svar:Datatypes.S dbi↦Y})
          rewrite 2!eval_mu_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
(let X0 := fresh_svar ϕ^{svar:Datatypes.S dbi↦X} in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0 (update_svar_val X S ρ)
        in
      eval ρ' ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X0})) =
(let X := fresh_svar ϕ^{svar:Datatypes.S dbi↦Y} in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X S0 (update_svar_val Y S ρ)
        in
      eval ρ' ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦X})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val
          (fresh_svar ϕ^{svar:Datatypes.S dbi↦X}) S0
          (update_svar_val X S ρ))
       ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh_svar
                                            ϕ^{svar:
                                            Datatypes.S
                                              dbi↦X}}) =
LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val
          (fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}) S0
          (update_svar_val Y S ρ))
       ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh_svar
                                            ϕ^{svar:
                                            Datatypes.S
                                              dbi↦Y}})
          apply f_equal.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
(λ S0 : propset M,
   eval
     (update_svar_val
        (fresh_svar ϕ^{svar:Datatypes.S dbi↦X}) S0
        (update_svar_val X S ρ))
     ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh_svar
                                          ϕ^{svar:
                                          Datatypes.S
                                            dbi↦X}}) =
(λ S0 : propset M,
   eval
     (update_svar_val
        (fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}) S0
        (update_svar_val Y S ρ))
     ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh_svar
                                          ϕ^{svar:
                                          Datatypes.S
                                            dbi↦Y}}) apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
∀ x : propset M,
  eval
    (update_svar_val
       (fresh_svar ϕ^{svar:Datatypes.S dbi↦X}) x
       (update_svar_val X S ρ))
    ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh_svar
                                         ϕ^{svar:Datatypes.S
                                                 dbi↦X}} =
  eval
    (update_svar_val
       (fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}) x
       (update_svar_val Y S ρ))
    ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh_svar
                                         ϕ^{svar:Datatypes.S
                                                 dbi↦Y}}
          intros S'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
eval
  (update_svar_val
     (fresh_svar ϕ^{svar:Datatypes.S dbi↦X}) S'
     (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh_svar
                                       ϕ^{svar:Datatypes.S
                                                 dbi↦X}} =
eval
  (update_svar_val
     (fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}) S'
     (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh_svar
                                       ϕ^{svar:Datatypes.S
                                                 dbi↦Y}}

          remember (fresh_svar (ϕ^{svar: Datatypes.S dbi ↦ X})) as X'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval
  (update_svar_val
     (fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}) S'
     (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh_svar
                                       ϕ^{svar:Datatypes.S
                                                 dbi↦Y}}
          remember (fresh_svar (ϕ^{svar: Datatypes.S dbi ↦ Y})) as Y'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          remember ((@singleton svar (@SVarSet signature) _ X)
                      ∪ ((singleton Y)
                           ∪ ((singleton X')
                                ∪ ((singleton Y')
                                     ∪ ((free_svars (ϕ^{svar: Datatypes.S dbi ↦ X}))
                                          ∪ (free_svars (ϕ^{svar: Datatypes.S dbi ↦ Y})
                                                        ∪ (free_svars ϕ))))))) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B: SVarSet
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))))
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          remember (svar_fresh (elements B)) as fresh3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B: SVarSet
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          assert(HB: fresh3 ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B: SVarSet
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
fresh3 ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B: SVarSet
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B: SVarSet
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
fresh3 ∉ B
            subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[fresh_svar ϕ^{svar:Datatypes.S dbi↦X}]}
            ∪ ({[fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}]}
               ∪ (free_svars
                    ϕ^{svar:Datatypes.S dbi↦X}
                  ∪ (free_svars
                       ϕ^{svar:Datatypes.S dbi↦Y}
                     ∪ free_svars ϕ)))))))
∉ {[X]}
  ∪ ({[Y]}
     ∪ ({[fresh_svar ϕ^{svar:Datatypes.S dbi↦X}]}
        ∪ ({[fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}]}
           ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
              ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
                 ∪ free_svars ϕ))))) apply set_svar_fresh_is_fresh'.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B: SVarSet
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}

          remember (free_svars (ϕ^{svar: Datatypes.S dbi ↦ Y}) ∪ (free_svars ϕ)) as B0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X} ∪ B0))))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          remember ((free_svars (ϕ^{svar: Datatypes.S dbi ↦ X})) ∪ B0) as B1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
HeqB: B = {[X]} ∪ ({[Y]} ∪ ({[X']} ∪ ({[Y']} ∪ B1)))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          remember ({[Y']} ∪ B1) as B2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
HeqB: B = {[X]} ∪ ({[Y]} ∪ ({[X']} ∪ B2))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          remember ({[X']} ∪ B2) as B3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
HeqB: B = {[X]} ∪ ({[Y]} ∪ B3)
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          remember ({[Y]} ∪ B3) as B4.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          pose proof (i := not_elem_of_union fresh3 {[X]} B4).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          pose proof (i0 := not_elem_of_union fresh3 {[Y]} B3).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[Y]} ∪ B3
↔ (fresh3 ∉ {[Y]}) ∧ fresh3 ∉ B3
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          pose proof (i1 := not_elem_of_union fresh3 {[X']} B2).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[Y]} ∪ B3
↔ (fresh3 ∉ {[Y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[X']} ∪ B2
↔ (fresh3 ∉ {[X']}) ∧ fresh3 ∉ B2
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          pose proof (i2 := not_elem_of_union fresh3 {[Y']} B1).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[Y]} ∪ B3
↔ (fresh3 ∉ {[Y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[X']} ∪ B2
↔ (fresh3 ∉ {[X']}) ∧ fresh3 ∉ B2
i2: fresh3 ∉ {[Y']} ∪ B1
↔ (fresh3 ∉ {[Y']}) ∧ fresh3 ∉ B1
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          pose proof (i3 := not_elem_of_union fresh3 
                                              (free_svars (ϕ^{svar: Datatypes.S dbi ↦ X})) 
                                              B0).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B0: SVarSet
HeqB0: B0 =
free_svars ϕ^{svar:Datatypes.S dbi↦Y}
∪ free_svars ϕ
B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[Y]} ∪ B3
↔ (fresh3 ∉ {[Y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[X']} ∪ B2
↔ (fresh3 ∉ {[X']}) ∧ fresh3 ∉ B2
i2: fresh3 ∉ {[Y']} ∪ B1
↔ (fresh3 ∉ {[Y']}) ∧ fresh3 ∉ B1
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X} ∪ B0
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3 ∉ B0
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          subst B0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B1: SVarSet
HeqB1: B1 =
free_svars ϕ^{svar:Datatypes.S dbi↦X}
∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
   ∪ free_svars ϕ)
B2: SVarSet
HeqB2: B2 = {[Y']} ∪ B1
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[Y]} ∪ B3
↔ (fresh3 ∉ {[Y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[X']} ∪ B2
↔ (fresh3 ∉ {[X']}) ∧ fresh3 ∉ B2
i2: fresh3 ∉ {[Y']} ∪ B1
↔ (fresh3 ∉ {[Y']}) ∧ fresh3 ∉ B1
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} subst B1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B2: SVarSet
HeqB2: B2 =
{[Y']}
∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
   ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ))
B3: SVarSet
HeqB3: B3 = {[X']} ∪ B2
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[Y]} ∪ B3
↔ (fresh3 ∉ {[Y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[X']} ∪ B2
↔ (fresh3 ∉ {[X']}) ∧ fresh3 ∉ B2
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} subst B2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B3: SVarSet
HeqB3: B3 =
{[X']}
∪ ({[Y']}
   ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)))
B4: SVarSet
HeqB4: B4 = {[Y]} ∪ B3
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[Y]} ∪ B3
↔ (fresh3 ∉ {[Y]}) ∧ fresh3 ∉ B3
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} subst B3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B, B4: SVarSet
HeqB4: B4 =
{[Y]}
∪ ({[X']}
   ∪ ({[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))))
HeqB: B = {[X]} ∪ B4
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[X]} ∪ B4
↔ (fresh3 ∉ {[X]}) ∧ fresh3 ∉ B4
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} subst B4.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
B: SVarSet
HeqB: B =
{[X]}
∪ ({[Y]}
   ∪ ({[X']}
      ∪ ({[Y']}
         ∪ (free_svars
              ϕ^{svar:
              Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))))
fresh3: svar
Heqfresh3: fresh3 = svar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3
∉ {[X]}
  ∪ ({[Y]}
     ∪ ({[X']}
        ∪ ({[Y']}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦X}
              ∪ (free_svars
                   ϕ^{svar:
                   Datatypes.S dbi↦Y}
                 ∪ free_svars ϕ)))))
↔ (fresh3 ∉ {[X]})
  ∧ fresh3
    ∉ {[Y]}
      ∪ ({[X']}
         ∪ ({[Y']}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦X}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦Y}
                  ∪ free_svars ϕ))))
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
HB: fresh3
∉ {[X]}
  ∪ ({[Y]}
     ∪ ({[X']}
        ∪ ({[Y']}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦X}
              ∪ (free_svars
                   ϕ^{svar:
                   Datatypes.S dbi↦Y}
                 ∪ free_svars ϕ)))))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i: fresh3
∉ {[X]}
  ∪ ({[Y]}
     ∪ ({[X']}
        ∪ ({[Y']}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦X}
              ∪ (free_svars
                   ϕ^{svar:
                   Datatypes.S dbi↦Y}
                 ∪ free_svars ϕ)))))
↔ (fresh3 ∉ {[X]})
  ∧ fresh3
    ∉ {[Y]}
      ∪ ({[X']}
         ∪ ({[Y']}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦X}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦Y}
                  ∪ free_svars ϕ))))
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          
          apply i in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
HB: (fresh3 ∉ {[X]})
∧ fresh3
  ∉ {[Y]}
    ∪ ({[X']}
       ∪ ({[Y']}
          ∪ (free_svars
               ϕ^{svar:
               Datatypes.S dbi↦X}
             ∪ (free_svars
                  ϕ^{svar:
                  Datatypes.S dbi↦Y}
                ∪ free_svars ϕ))))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i: fresh3
∉ {[X]}
  ∪ ({[Y]}
     ∪ ({[X']}
        ∪ ({[Y']}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦X}
              ∪ (free_svars
                   ϕ^{svar:
                   Datatypes.S dbi↦Y}
                 ∪ free_svars ϕ)))))
↔ (fresh3 ∉ {[X]})
  ∧ fresh3
    ∉ {[Y]}
      ∪ ({[X']}
         ∪ ({[Y']}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦X}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦Y}
                  ∪ free_svars ϕ))))
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} clear i.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
HB: (fresh3 ∉ {[X]})
∧ fresh3
  ∉ {[Y]}
    ∪ ({[X']}
       ∪ ({[Y']}
          ∪ (free_svars
               ϕ^{svar:
               Datatypes.S dbi↦X}
             ∪ (free_svars
                  ϕ^{svar:
                  Datatypes.S dbi↦Y}
                ∪ free_svars ϕ))))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} destruct HB as [Hneqfr1 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ∉ {[X]}
HB: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}        
          apply not_elem_of_singleton_1 in Hneqfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
HB: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}

          apply i0 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
HB: (fresh3 ∉ {[Y]})
∧ fresh3
  ∉ {[X']}
    ∪ ({[Y']}
       ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
          ∪ (free_svars
               ϕ^{svar:
               Datatypes.S dbi↦Y} ∪ 
             free_svars ϕ)))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i0: fresh3
∉ {[Y]}
  ∪ ({[X']}
     ∪ ({[Y']}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
           ∪ (free_svars
                ϕ^{svar:
                Datatypes.S dbi↦Y} ∪ 
              free_svars ϕ))))
↔ (fresh3 ∉ {[Y]})
  ∧ fresh3
    ∉ {[X']}
      ∪ ({[Y']}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
            ∪ (free_svars
                 ϕ^{svar:
                 Datatypes.S dbi↦Y}
               ∪ free_svars ϕ)))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} clear i0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
HB: (fresh3 ∉ {[Y]})
∧ fresh3
  ∉ {[X']}
    ∪ ({[Y']}
       ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
          ∪ (free_svars
               ϕ^{svar:
               Datatypes.S dbi↦Y} ∪ 
             free_svars ϕ)))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} destruct HB as [Hneqfr2 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ∉ {[Y]}
HB: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          apply not_elem_of_singleton_1 in Hneqfr2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
HB: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}

          apply i1 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
HB: (fresh3 ∉ {[X']})
∧ fresh3
  ∉ {[Y']}
    ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
       ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
          ∪ free_svars ϕ))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i1: fresh3
∉ {[X']}
  ∪ ({[Y']}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
        ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
           ∪ free_svars ϕ)))
↔ (fresh3 ∉ {[X']})
  ∧ fresh3
    ∉ {[Y']}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
         ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
            ∪ free_svars ϕ))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} clear i1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
HB: (fresh3 ∉ {[X']})
∧ fresh3
  ∉ {[Y']}
    ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
       ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
          ∪ free_svars ϕ))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} destruct HB as [Hneqfr11 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ∉ {[X']}
HB: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          apply not_elem_of_singleton_1 in Hneqfr11.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
HB: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}

          apply i2 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
HB: (fresh3 ∉ {[Y']})
∧ fresh3
  ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
    ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
       ∪ free_svars ϕ)
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i2: fresh3
∉ {[Y']}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦X}
     ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
        ∪ free_svars ϕ))
↔ (fresh3 ∉ {[Y']})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
      ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
         ∪ free_svars ϕ)
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} clear i2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
HB: (fresh3 ∉ {[Y']})
∧ fresh3
  ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
    ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
       ∪ free_svars ϕ)
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} destruct HB as [Hneqfr22 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ∉ {[Y']}
HB: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          apply not_elem_of_singleton_1 in Hneqfr22.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HB: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}

          apply i3 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HB: (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
∧ fresh3
  ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
    ∪ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
i3: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
  ∪ (free_svars ϕ^{svar:Datatypes.S dbi↦Y}
     ∪ free_svars ϕ)
↔ (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
  ∧ fresh3
    ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
      ∪ free_svars ϕ
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} clear i3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HB: (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X})
∧ fresh3
  ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
    ∪ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} destruct HB as [HnotinFree1 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HB: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
  ∪ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          apply not_elem_of_union in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HB: (fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y})
∧ fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          destruct HB as [HnotinFree2 HnotinFree].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          fold bsvar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval (update_svar_val X' S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦X'} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}

          rewrite (proj1 (IHsz _ _ _ _) X' fresh3).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
size ϕ^{svar:Datatypes.S dbi↦X} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in X' ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} rewrite -svar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in X' ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in X' ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          rewrite HeqX'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in
  (fresh_svar ϕ^{svar:Datatypes.S dbi↦X})
  ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} apply set_svar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} unfold svar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'} apply HnotinFree1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval (update_svar_val Y' S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦Y'}
          rewrite (proj1 (IHsz _ _ _ _) Y' fresh3).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
size ϕ^{svar:Datatypes.S dbi↦Y} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y' ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3} rewrite -svar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y' ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y' ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3}
          rewrite HeqY'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in
  (fresh_svar ϕ^{svar:Datatypes.S dbi↦Y})
  ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3} apply set_svar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in fresh3 ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3} unfold svar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3} apply HnotinFree2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:Datatypes.S dbi↦X}^{svar:0↦fresh3} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3}
          rewrite (svar_open_comm_higher 0 (Datatypes.S dbi) _ fresh3 X ϕ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
0 < Datatypes.S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:0↦fresh3}^{svar:
  Init.Nat.pred (Datatypes.S dbi)↦X} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:0↦fresh3}^{svar:Init.Nat.pred
                            (Datatypes.S dbi)↦X} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:Datatypes.S dbi↦Y}^{svar:0↦fresh3}
          rewrite (svar_open_comm_higher 0 (Datatypes.S dbi) _ fresh3 Y ϕ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
0 < Datatypes.S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:0↦fresh3}^{svar:
  Init.Nat.pred (Datatypes.S dbi)↦X} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:0↦fresh3}^{svar:
  Init.Nat.pred (Datatypes.S dbi)↦Y} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:0↦fresh3}^{svar:Init.Nat.pred
                            (Datatypes.S dbi)↦X} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:0↦fresh3}^{svar:Init.Nat.pred
                            (Datatypes.S dbi)↦Y}
          replace (pred (Datatypes.S dbi)) with dbi by lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val fresh3 S' (update_svar_val X S ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦X} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          rewrite (update_svar_val_comm fresh3 X).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val X S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦X} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} apply Hneqfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val X S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦X} =
eval
  (update_svar_val fresh3 S' (update_svar_val Y S ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          rewrite (update_svar_val_comm fresh3 Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ≠ Y
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val X S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦X} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} apply Hneqfr2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (mu , ϕ)
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val X S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦X} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          apply svar_is_fresh_in_exists in HfrX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (mu , ϕ)
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val X S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦X} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          apply svar_is_fresh_in_exists in HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val X S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦X} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          rewrite (proj1 (IHsz _ _ _ _) X Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
size ϕ^{svar:0↦fresh3} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in X ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} rewrite -svar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in X ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in X ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          unfold svar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
X ∉ free_svars ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          apply svar_open_fresh_notin.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
X ∉ free_svars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ∉ free_svars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
X ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          apply HfrX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ∉ free_svars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
X ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} apply HnotinFree.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
X ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} intros Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Contra: X = fresh3
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} symmetry in Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Contra: fresh3 = X
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} contradiction.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
svar_is_fresh_in Y ϕ^{svar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          apply svar_open_fresh_notin.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Y ∉ free_svars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ∉ free_svars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Y ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          apply HfrY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
fresh3 ∉ free_svars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Y ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} apply HnotinFree.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Y ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} intros Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Contra: Y = fresh3
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} symmetry in Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
Contra: fresh3 = Y
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} contradiction.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: Datatypes.S (size ϕ) ≤ Datatypes.S sz
X, Y: svar
S: propset M
HfrX: svar_is_fresh_in X (ex , (mu , ϕ))
HfrY: svar_is_fresh_in Y (ex , (mu , ϕ))
S': propset M
X': svar
HeqX': X' = fresh_svar ϕ^{svar:Datatypes.S dbi↦X}
Y': svar
HeqY': Y' = fresh_svar ϕ^{svar:Datatypes.S dbi↦Y}
fresh3: svar
Hneqfr1: fresh3 ≠ X
Hneqfr2: fresh3 ≠ Y
Hneqfr11: fresh3 ≠ X'
Hneqfr22: fresh3 ≠ Y'
HnotinFree1: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦X}
HnotinFree2: fresh3
∉ free_svars ϕ^{svar:Datatypes.S dbi↦Y}
HnotinFree: fresh3 ∉ free_svars ϕ
Heqfresh3: fresh3 =
svar_fresh
  (elements
     ({[X]}
      ∪ ({[Y]}
         ∪ ({[X']}
            ∪ ({[Y']}
               ∪ (free_svars
                    ϕ^{svar:
                    Datatypes.S dbi↦X}
                  ∪ (free_svars
                     ϕ^{svar:
                     Datatypes.S dbi↦Y}
                     ∪ 
                     free_svars ϕ)))))))
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y} =
eval
  (update_svar_val Y S (update_svar_val fresh3 S' ρ))
  ϕ^{svar:0↦fresh3}^{svar:dbi↦Y}
          reflexivity.
      }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ S sz
∀ (x y : evar) (c : M),
  evar_is_fresh_in x ϕ
  → evar_is_fresh_in y ϕ
    → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
      eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
      {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ S sz
∀ (x y : evar) (c : M),
  evar_is_fresh_in x ϕ
  → evar_is_fresh_in y ϕ
    → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
      eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
        (* inductive case - evar *)
        move=> x y c Hfrx Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: size ϕ ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ϕ
Hfry: evar_is_fresh_in y ϕ
eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
        destruct ϕ; simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
eval (update_evar_val x c ρ)
  (patt_free_evar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_evar x0)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
eval (update_evar_val x c ρ)
  (patt_free_svar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_svar x0)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
eval (update_evar_val x c ρ)
  (patt_bound_evar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_evar n)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
eval (update_evar_val x c ρ)
  (patt_bound_svar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_svar n)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
eval (update_evar_val x c ρ)
  (patt_sym sigma)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_sym sigma)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
eval (update_evar_val x c ρ) (ϕ1 $ ϕ2)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (ϕ1 $ ϕ2)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
eval (update_evar_val x c ρ) ⊥^{evar:dbi↦x} =
eval (update_evar_val y c ρ) ⊥^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
eval (update_evar_val x c ρ) (ϕ1 ---> ϕ2)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (ϕ1 ---> ϕ2)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
eval (update_evar_val x c ρ) (ex , ϕ)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (ex , ϕ)^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
eval (update_evar_val x c ρ) (mu , ϕ)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (mu , ϕ)^{evar:dbi↦y}
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
eval (update_evar_val x c ρ)
  (patt_free_evar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_evar x0)^{evar:dbi↦y} rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
size (patt_free_evar x0) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
evar_is_fresh_in x (patt_free_evar x0)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
evar_is_fresh_in y (patt_free_evar x0)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
eval (update_evar_val y c ρ)
  (patt_free_evar x0)^{evar:dbi↦y} =
eval (update_evar_val y c ρ)
  (patt_free_evar x0)^{evar:dbi↦y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
size (patt_free_evar x0) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
evar_is_fresh_in x (patt_free_evar x0)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
evar_is_fresh_in y (patt_free_evar x0) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
size (patt_free_evar x0) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: evar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_evar x0)
Hfry: evar_is_fresh_in y (patt_free_evar x0)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
eval (update_evar_val x c ρ)
  (patt_free_svar x0)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_free_svar x0)^{evar:dbi↦y} rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
size (patt_free_svar x0) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
evar_is_fresh_in x (patt_free_svar x0)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
evar_is_fresh_in y (patt_free_svar x0)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
eval (update_evar_val y c ρ)
  (patt_free_svar x0)^{evar:dbi↦y} =
eval (update_evar_val y c ρ)
  (patt_free_svar x0)^{evar:dbi↦y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
size (patt_free_svar x0) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
evar_is_fresh_in x (patt_free_svar x0)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
evar_is_fresh_in y (patt_free_svar x0) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
size (patt_free_svar x0) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
x0: svar
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_free_svar x0)
Hfry: evar_is_fresh_in y (patt_free_svar x0)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
eval (update_evar_val x c ρ)
  (patt_bound_evar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_evar n)^{evar:dbi↦y} rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
size (patt_bound_evar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
evar_is_fresh_in x (patt_bound_evar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
evar_is_fresh_in y (patt_bound_evar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
eval (update_evar_val y c ρ)
  (patt_bound_evar n)^{evar:dbi↦y} =
eval (update_evar_val y c ρ)
  (patt_bound_evar n)^{evar:dbi↦y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
size (patt_bound_evar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
evar_is_fresh_in x (patt_bound_evar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
evar_is_fresh_in y (patt_bound_evar n) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
size (patt_bound_evar n) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_evar n)
Hfry: evar_is_fresh_in y (patt_bound_evar n)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
eval (update_evar_val x c ρ)
  (patt_bound_svar n)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_bound_svar n)^{evar:dbi↦y} rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
size (patt_bound_svar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
evar_is_fresh_in x (patt_bound_svar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
evar_is_fresh_in y (patt_bound_svar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
eval (update_evar_val y c ρ)
  (patt_bound_svar n)^{evar:dbi↦y} =
eval (update_evar_val y c ρ)
  (patt_bound_svar n)^{evar:dbi↦y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
size (patt_bound_svar n) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
evar_is_fresh_in x (patt_bound_svar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
evar_is_fresh_in y (patt_bound_svar n) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
size (patt_bound_svar n) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
n, dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_bound_svar n)
Hfry: evar_is_fresh_in y (patt_bound_svar n)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
eval (update_evar_val x c ρ)
  (patt_sym sigma)^{evar:dbi↦x} =
eval (update_evar_val y c ρ)
  (patt_sym sigma)^{evar:dbi↦y} rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
size (patt_sym sigma) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
evar_is_fresh_in x (patt_sym sigma)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
evar_is_fresh_in y (patt_sym sigma)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
eval (update_evar_val y c ρ)
  (patt_sym sigma)^{evar:dbi↦y} =
eval (update_evar_val y c ρ)
  (patt_sym sigma)^{evar:dbi↦y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
size (patt_sym sigma) ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
evar_is_fresh_in x (patt_sym sigma)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
evar_is_fresh_in y (patt_sym sigma) 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
size (patt_sym sigma) ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
sigma: symbols
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (patt_sym sigma)
Hfry: evar_is_fresh_in y (patt_sym sigma)
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
eval (update_evar_val x c ρ) (ϕ1 $ ϕ2)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (ϕ1 $ ϕ2)^{evar:dbi↦y} rewrite 2!eval_app_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext
  (eval (update_evar_val x c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ1))
  (eval (update_evar_val x c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ2)) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2)) fold evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext
  (eval (update_evar_val x c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ1))
  (eval (update_evar_val x c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ2)) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2))
          rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val x c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ2)) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2)) 4: rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2)) lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2))
          eapply evar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2)) apply Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2))
          eapply evar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2)) apply Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2))
          lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2))
          eapply evar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in x (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2)) apply Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2))
          eapply evar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
evar_is_fresh_in y (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) 
         (x : db_index) 
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2)) apply Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 $ ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 $ ϕ2)
app_ext (eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y})
  (eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y}) =
app_ext
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ1))
  (eval (update_evar_val y c ρ)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ2))
          reflexivity.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
eval (update_evar_val x c ρ) ⊥^{evar:dbi↦x} =
eval (update_evar_val y c ρ) ⊥^{evar:dbi↦y} rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
size ⊥ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
evar_is_fresh_in x ⊥
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
evar_is_fresh_in y ⊥
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
eval (update_evar_val y c ρ) ⊥^{evar:dbi↦y} =
eval (update_evar_val y c ρ) ⊥^{evar:dbi↦y} 4: reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
size ⊥ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
evar_is_fresh_in x ⊥
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
evar_is_fresh_in y ⊥ 3,2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
size ⊥ ≤ sz simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
dbi: db_index
ρ: Valuation
Hsz: 0 ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x ⊥
Hfry: evar_is_fresh_in y ⊥
0 ≤ sz lia.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
eval (update_evar_val x c ρ) (ϕ1 ---> ϕ2)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (ϕ1 ---> ϕ2)^{evar:dbi↦y} rewrite 2!eval_imp_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤
∖ eval (update_evar_val x c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar x) dbi ϕ1)
∪ eval (update_evar_val x c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar x) dbi ϕ2) =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2) fold evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤
∖ eval (update_evar_val x c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar x) dbi ϕ1)
∪ eval (update_evar_val x c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar x) dbi ϕ2) =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2)
          rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val x c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar x) dbi ϕ2) =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2) 4: rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2) lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2)
          eapply evar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2) apply Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2)
          eapply evar_is_fresh_in_app_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y (ϕ1 $ ?ϕ₂)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2) apply Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
size ϕ2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2)
          lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2)
          eapply evar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in x (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2) apply Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y ϕ2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2)
          eapply evar_is_fresh_in_app_r.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
evar_is_fresh_in y (?ϕ₁ $ ϕ2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) 
        (x : db_index) 
        (phi : Pattern) {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2) apply Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ1, ϕ2: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ1 + size ϕ2) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ϕ1 ---> ϕ2)
Hfry: evar_is_fresh_in y (ϕ1 ---> ϕ2)
⊤ ∖ eval (update_evar_val y c ρ) ϕ1^{evar:dbi↦y}
∪ eval (update_evar_val y c ρ) ϕ2^{evar:dbi↦y} =
⊤
∖ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ1)
∪ eval (update_evar_val y c ρ)
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ2)
          reflexivity.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
eval (update_evar_val x c ρ) (ex , ϕ)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (ex , ϕ)^{evar:dbi↦y} mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
eval (update_evar_val x c ρ) (ex , ϕ^{evar:S dbi↦x}) =
eval (update_evar_val y c ρ)
  (ex , ϕ^{evar:increase_ex pm_spec_data dbi↦y}) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
eval (update_evar_val x c ρ) (ex , ϕ^{evar:S dbi↦x}) =
eval (update_evar_val y c ρ) (ex , ϕ^{evar:S dbi↦y})
          rewrite 2!eval_ex_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
(let x0 := fresh_evar ϕ^{evar:S dbi↦x} in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x0 e (update_evar_val x c ρ)
        in
      eval ρ' ϕ^{evar:S dbi↦x}^{evar:0↦x0})) =
(let x := fresh_evar ϕ^{evar:S dbi↦y} in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_evar_val y c ρ) in
      eval ρ' ϕ^{evar:S dbi↦y}^{evar:0↦x})) fold evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
(let x0 := fresh_evar ϕ^{evar:S dbi↦x} in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x0 e (update_evar_val x c ρ)
        in
      eval ρ' ϕ^{evar:S dbi↦x}^{evar:0↦x0})) =
(let x := fresh_evar ϕ^{evar:S dbi↦y} in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e (update_evar_val y c ρ) in
      eval ρ' ϕ^{evar:S dbi↦y}^{evar:0↦x})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val (fresh_evar ϕ^{evar:S dbi↦x})
          e (update_evar_val x c ρ))
       ϕ^{evar:S dbi↦x}^{evar:0↦fresh_evar
                                  ϕ^{evar:S dbi↦x}}) =
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val (fresh_evar ϕ^{evar:S dbi↦y})
          e (update_evar_val y c ρ))
       ϕ^{evar:S dbi↦y}^{evar:0↦fresh_evar
                                  ϕ^{evar:S dbi↦y}})
          apply f_equal.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
(λ e : M,
   eval
     (update_evar_val (fresh_evar ϕ^{evar:S dbi↦x}) e
        (update_evar_val x c ρ))
     ϕ^{evar:S dbi↦x}^{evar:0↦fresh_evar
                                ϕ^{evar:S dbi↦x}}) =
(λ e : M,
   eval
     (update_evar_val (fresh_evar ϕ^{evar:S dbi↦y}) e
        (update_evar_val y c ρ))
     ϕ^{evar:S dbi↦y}^{evar:0↦fresh_evar
                                ϕ^{evar:S dbi↦y}}) apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
∀ x0 : M,
  eval
    (update_evar_val (fresh_evar ϕ^{evar:S dbi↦x}) x0
       (update_evar_val x c ρ))
    ϕ^{evar:S dbi↦x}^{evar:0↦fresh_evar
                               ϕ^{evar:S dbi↦x}} =
  eval
    (update_evar_val (fresh_evar ϕ^{evar:S dbi↦y}) x0
       (update_evar_val y c ρ))
    ϕ^{evar:S dbi↦y}^{evar:0↦fresh_evar
                               ϕ^{evar:S dbi↦y}} intros e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
eval
  (update_evar_val (fresh_evar ϕ^{evar:S dbi↦x}) e
     (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh_evar ϕ^{evar:S dbi↦x}} =
eval
  (update_evar_val (fresh_evar ϕ^{evar:S dbi↦y}) e
     (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh_evar ϕ^{evar:S dbi↦y}}
          remember (fresh_evar (ϕ^{evar: Datatypes.S dbi ↦ x})) as x'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval
  (update_evar_val (fresh_evar ϕ^{evar:S dbi↦y}) e
     (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh_evar ϕ^{evar:S dbi↦y}}
          remember (fresh_evar (ϕ^{evar: Datatypes.S dbi ↦ y})) as y'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          remember ((@singleton evar (@EVarSet signature) _ x)
                      ∪ ((singleton y)
                           ∪ ((singleton x')
                                ∪ ((singleton y')
                                     ∪ ((free_evars (ϕ^{evar: Datatypes.S dbi ↦ x}))
                                          ∪ (free_evars (ϕ^{evar: Datatypes.S dbi ↦ y})
                                                        ∪ (free_evars ϕ))))))) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B: EVarSet
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))))
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          remember (evar_fresh (elements B)) as fresh3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B: EVarSet
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          assert(HB: fresh3 ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B: EVarSet
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
fresh3 ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B: EVarSet
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B: EVarSet
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
fresh3 ∉ B
            subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[fresh_evar ϕ^{evar:S dbi↦x}]}
            ∪ ({[fresh_evar ϕ^{evar:S dbi↦y}]}
               ∪ (free_evars ϕ^{evar:S dbi↦x}
                  ∪ (free_evars ϕ^{evar:S dbi↦y}
                     ∪ free_evars ϕ)))))))
∉ {[x]}
  ∪ ({[y]}
     ∪ ({[fresh_evar ϕ^{evar:S dbi↦x}]}
        ∪ ({[fresh_evar ϕ^{evar:S dbi↦y}]}
           ∪ (free_evars ϕ^{evar:S dbi↦x}
              ∪ (free_evars ϕ^{evar:S dbi↦y}
                 ∪ free_evars ϕ))))) apply set_evar_fresh_is_fresh'.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B: EVarSet
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}

          remember (free_evars (ϕ^{evar: Datatypes.S dbi ↦ y}) ∪ (free_evars ϕ)) as B0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x} ∪ B0))))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          remember ((free_evars (ϕ^{evar: Datatypes.S dbi ↦ x})) ∪ B0) as B1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
HeqB: B = {[x]} ∪ ({[y]} ∪ ({[x']} ∪ ({[y']} ∪ B1)))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          remember ({[y']} ∪ B1) as B2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
HeqB: B = {[x]} ∪ ({[y]} ∪ ({[x']} ∪ B2))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          remember ({[x']} ∪ B2) as B3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
HeqB: B = {[x]} ∪ ({[y]} ∪ B3)
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          remember ({[y]} ∪ B3) as B4.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          pose proof (i := not_elem_of_union fresh3 {[x]} B4).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          pose proof (i0 := not_elem_of_union fresh3 {[y]} B3).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[y]} ∪ B3
↔ (fresh3 ∉ {[y]}) ∧ fresh3 ∉ B3
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          pose proof (i1 := not_elem_of_union fresh3 {[x']} B2).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[y]} ∪ B3
↔ (fresh3 ∉ {[y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[x']} ∪ B2
↔ (fresh3 ∉ {[x']}) ∧ fresh3 ∉ B2
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          pose proof (i2 := not_elem_of_union fresh3 {[y']} B1).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[y]} ∪ B3
↔ (fresh3 ∉ {[y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[x']} ∪ B2
↔ (fresh3 ∉ {[x']}) ∧ fresh3 ∉ B2
i2: fresh3 ∉ {[y']} ∪ B1
↔ (fresh3 ∉ {[y']}) ∧ fresh3 ∉ B1
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          pose proof (i3 := not_elem_of_union fresh3 
                                              (free_evars (ϕ^{evar: Datatypes.S dbi ↦ x})) 
                                              B0).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B0: EVarSet
HeqB0: B0 =
free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
B1: EVarSet
HeqB1: B1 = free_evars ϕ^{evar:S dbi↦x} ∪ B0
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[y]} ∪ B3
↔ (fresh3 ∉ {[y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[x']} ∪ B2
↔ (fresh3 ∉ {[x']}) ∧ fresh3 ∉ B2
i2: fresh3 ∉ {[y']} ∪ B1
↔ (fresh3 ∉ {[y']}) ∧ fresh3 ∉ B1
i3: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x} ∪ B0
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3 ∉ B0
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          subst B0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B1: EVarSet
HeqB1: B1 =
free_evars ϕ^{evar:S dbi↦x}
∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
B2: EVarSet
HeqB2: B2 = {[y']} ∪ B1
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[y]} ∪ B3
↔ (fresh3 ∉ {[y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[x']} ∪ B2
↔ (fresh3 ∉ {[x']}) ∧ fresh3 ∉ B2
i2: fresh3 ∉ {[y']} ∪ B1
↔ (fresh3 ∉ {[y']}) ∧ fresh3 ∉ B1
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} subst B1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B2: EVarSet
HeqB2: B2 =
{[y']}
∪ (free_evars ϕ^{evar:S dbi↦x}
   ∪ (free_evars ϕ^{evar:S dbi↦y}
      ∪ free_evars ϕ))
B3: EVarSet
HeqB3: B3 = {[x']} ∪ B2
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[y]} ∪ B3
↔ (fresh3 ∉ {[y]}) ∧ fresh3 ∉ B3
i1: fresh3 ∉ {[x']} ∪ B2
↔ (fresh3 ∉ {[x']}) ∧ fresh3 ∉ B2
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} subst B2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B3: EVarSet
HeqB3: B3 =
{[x']}
∪ ({[y']}
   ∪ (free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)))
B4: EVarSet
HeqB4: B4 = {[y]} ∪ B3
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3 ∉ {[y]} ∪ B3
↔ (fresh3 ∉ {[y]}) ∧ fresh3 ∉ B3
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} subst B3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B, B4: EVarSet
HeqB4: B4 =
{[y]}
∪ ({[x']}
   ∪ ({[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))))
HeqB: B = {[x]} ∪ B4
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3 ∉ {[x]} ∪ B4
↔ (fresh3 ∉ {[x]}) ∧ fresh3 ∉ B4
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} subst B4.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
B: EVarSet
HeqB: B =
{[x]}
∪ ({[y]}
   ∪ ({[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))))
fresh3: evar
Heqfresh3: fresh3 = evar_fresh (elements B)
HB: fresh3 ∉ B
i: fresh3
∉ {[x]}
  ∪ ({[y]}
     ∪ ({[x']}
        ∪ ({[y']}
           ∪ (free_evars ϕ^{evar:S dbi↦x}
              ∪ (free_evars ϕ^{evar:S dbi↦y}
                 ∪ free_evars ϕ)))))
↔ (fresh3 ∉ {[x]})
  ∧ fresh3
    ∉ {[y]}
      ∪ ({[x']}
         ∪ ({[y']}
            ∪ (free_evars ϕ^{evar:S dbi↦x}
               ∪ (free_evars ϕ^{evar:S dbi↦y}
                  ∪ free_evars ϕ))))
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
HB: fresh3
∉ {[x]}
  ∪ ({[y]}
     ∪ ({[x']}
        ∪ ({[y']}
           ∪ (free_evars ϕ^{evar:S dbi↦x}
              ∪ (free_evars ϕ^{evar:S dbi↦y}
                 ∪ free_evars ϕ)))))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i: fresh3
∉ {[x]}
  ∪ ({[y]}
     ∪ ({[x']}
        ∪ ({[y']}
           ∪ (free_evars ϕ^{evar:S dbi↦x}
              ∪ (free_evars ϕ^{evar:S dbi↦y}
                 ∪ free_evars ϕ)))))
↔ (fresh3 ∉ {[x]})
  ∧ fresh3
    ∉ {[y]}
      ∪ ({[x']}
         ∪ ({[y']}
            ∪ (free_evars ϕ^{evar:S dbi↦x}
               ∪ (free_evars ϕ^{evar:S dbi↦y}
                  ∪ free_evars ϕ))))
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}

          apply i in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
HB: (fresh3 ∉ {[x]})
∧ fresh3
  ∉ {[y]}
    ∪ ({[x']}
       ∪ ({[y']}
          ∪ (free_evars ϕ^{evar:S dbi↦x}
             ∪ (free_evars ϕ^{evar:S dbi↦y}
                ∪ free_evars ϕ))))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i: fresh3
∉ {[x]}
  ∪ ({[y]}
     ∪ ({[x']}
        ∪ ({[y']}
           ∪ (free_evars ϕ^{evar:S dbi↦x}
              ∪ (free_evars ϕ^{evar:S dbi↦y}
                 ∪ free_evars ϕ)))))
↔ (fresh3 ∉ {[x]})
  ∧ fresh3
    ∉ {[y]}
      ∪ ({[x']}
         ∪ ({[y']}
            ∪ (free_evars ϕ^{evar:S dbi↦x}
               ∪ (free_evars ϕ^{evar:S dbi↦y}
                  ∪ free_evars ϕ))))
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} clear i.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
HB: (fresh3 ∉ {[x]})
∧ fresh3
  ∉ {[y]}
    ∪ ({[x']}
       ∪ ({[y']}
          ∪ (free_evars ϕ^{evar:S dbi↦x}
             ∪ (free_evars ϕ^{evar:S dbi↦y}
                ∪ free_evars ϕ))))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} destruct HB as [Hneqfr1 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ∉ {[x]}
HB: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          apply not_elem_of_singleton_1 in Hneqfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
HB: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}

          apply i0 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
HB: (fresh3 ∉ {[y]})
∧ fresh3
  ∉ {[x']}
    ∪ ({[y']}
       ∪ (free_evars ϕ^{evar:S dbi↦x}
          ∪ (free_evars ϕ^{evar:S dbi↦y}
             ∪ free_evars ϕ)))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i0: fresh3
∉ {[y]}
  ∪ ({[x']}
     ∪ ({[y']}
        ∪ (free_evars ϕ^{evar:S dbi↦x}
           ∪ (free_evars ϕ^{evar:S dbi↦y}
              ∪ free_evars ϕ))))
↔ (fresh3 ∉ {[y]})
  ∧ fresh3
    ∉ {[x']}
      ∪ ({[y']}
         ∪ (free_evars ϕ^{evar:S dbi↦x}
            ∪ (free_evars ϕ^{evar:S dbi↦y}
               ∪ free_evars ϕ)))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} clear i0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
HB: (fresh3 ∉ {[y]})
∧ fresh3
  ∉ {[x']}
    ∪ ({[y']}
       ∪ (free_evars ϕ^{evar:S dbi↦x}
          ∪ (free_evars ϕ^{evar:S dbi↦y}
             ∪ free_evars ϕ)))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} destruct HB as [Hneqfr2 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ∉ {[y]}
HB: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          apply not_elem_of_singleton_1 in Hneqfr2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
HB: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}

          apply i1 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
HB: (fresh3 ∉ {[x']})
∧ fresh3
  ∉ {[y']}
    ∪ (free_evars ϕ^{evar:S dbi↦x}
       ∪ (free_evars ϕ^{evar:S dbi↦y}
          ∪ free_evars ϕ))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i1: fresh3
∉ {[x']}
  ∪ ({[y']}
     ∪ (free_evars ϕ^{evar:S dbi↦x}
        ∪ (free_evars ϕ^{evar:S dbi↦y}
           ∪ free_evars ϕ)))
↔ (fresh3 ∉ {[x']})
  ∧ fresh3
    ∉ {[y']}
      ∪ (free_evars ϕ^{evar:S dbi↦x}
         ∪ (free_evars ϕ^{evar:S dbi↦y}
            ∪ free_evars ϕ))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} clear i1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
HB: (fresh3 ∉ {[x']})
∧ fresh3
  ∉ {[y']}
    ∪ (free_evars ϕ^{evar:S dbi↦x}
       ∪ (free_evars ϕ^{evar:S dbi↦y}
          ∪ free_evars ϕ))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} destruct HB as [Hneqfr11 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ∉ {[x']}
HB: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          apply not_elem_of_singleton_1 in Hneqfr11.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
HB: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}

          apply i2 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
HB: (fresh3 ∉ {[y']})
∧ fresh3
  ∉ free_evars ϕ^{evar:S dbi↦x}
    ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i2: fresh3
∉ {[y']}
  ∪ (free_evars ϕ^{evar:S dbi↦x}
     ∪ (free_evars ϕ^{evar:S dbi↦y}
        ∪ free_evars ϕ))
↔ (fresh3 ∉ {[y']})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦x}
      ∪ (free_evars ϕ^{evar:S dbi↦y}
         ∪ free_evars ϕ)
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} clear i2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
HB: (fresh3 ∉ {[y']})
∧ fresh3
  ∉ free_evars ϕ^{evar:S dbi↦x}
    ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} destruct HB as [Hneqfr22 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ∉ {[y']}
HB: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          apply not_elem_of_singleton_1 in Hneqfr22.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HB: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}

          apply i3 in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HB: (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
∧ fresh3
  ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
i3: fresh3
∉ free_evars ϕ^{evar:S dbi↦x}
  ∪ (free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ)
↔ (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
  ∧ fresh3
    ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} clear i3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HB: (fresh3 ∉ free_evars ϕ^{evar:S dbi↦x})
∧ fresh3
  ∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} destruct HB as [HnotinFree1 HB].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HB: fresh3
∉ free_evars ϕ^{evar:S dbi↦y} ∪ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          apply not_elem_of_union in HB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HB: (fresh3 ∉ free_evars ϕ^{evar:S dbi↦y})
∧ fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          destruct HB as [HnotinFree2 HnotinFree].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval (update_evar_val x' e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦x'} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          rewrite (proj2 (IHsz _ _ _ _) x' fresh3).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
size ϕ^{evar:S dbi↦x} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in x' ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} rewrite -evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in x' ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in x' ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          rewrite Heqx'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in (fresh_evar ϕ^{evar:S dbi↦x})
  ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'} apply HnotinFree1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval (update_evar_val y' e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦y'}
          rewrite (proj2 (IHsz _ _ _ _) y' fresh3).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
size ϕ^{evar:S dbi↦y} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y' ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3} rewrite -evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y' ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y' ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3}
          rewrite Heqy'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in (fresh_evar ϕ^{evar:S dbi↦y})
  ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3} apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in fresh3 ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3} unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3} apply HnotinFree2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:S dbi↦x}^{evar:0↦fresh3} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3}
          rewrite (evar_open_comm_higher 0 (Datatypes.S dbi) _ fresh3 x ϕ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:0↦fresh3}^{evar:
  Init.Nat.pred (S dbi)↦x} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:0↦fresh3}^{evar:Init.Nat.pred (S dbi)↦x} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:S dbi↦y}^{evar:0↦fresh3}
          rewrite (evar_open_comm_higher 0 (Datatypes.S dbi) _ fresh3 y ϕ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:0↦fresh3}^{evar:
  Init.Nat.pred (S dbi)↦x} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:0↦fresh3}^{evar:
  Init.Nat.pred (S dbi)↦y} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:0↦fresh3}^{evar:Init.Nat.pred (S dbi)↦x} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:0↦fresh3}^{evar:Init.Nat.pred (S dbi)↦y}
          replace (pred (Datatypes.S dbi)) with dbi by lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val fresh3 e (update_evar_val x c ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦x} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          rewrite (update_evar_val_comm fresh3 x).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ≠ x
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val x c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦x} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} apply Hneqfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val x c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦x} =
eval
  (update_evar_val fresh3 e (update_evar_val y c ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          rewrite (update_evar_val_comm fresh3 y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ≠ y
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val x c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦x} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} apply Hneqfr2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , ϕ)
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val x c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦x} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          apply evar_is_fresh_in_exists in Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , ϕ)
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val x c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦x} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          apply evar_is_fresh_in_exists in Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val x c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦x} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
size ϕ^{evar:0↦fresh3} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in x ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} rewrite -evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in x ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in x ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
x ∉ free_evars ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          apply evar_open_fresh_notin.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
x ∉ free_evars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ∉ free_evars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
x ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          apply Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ∉ free_evars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
x ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} apply HnotinFree.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
x ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} intros Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
Contra: x = fresh3
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} symmetry in Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
Contra: fresh3 = x
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} contradiction.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
evar_is_fresh_in y ϕ^{evar:0↦fresh3}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          apply evar_open_fresh_notin.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
y ∉ free_evars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ∉ free_evars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
y ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          apply Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
fresh3 ∉ free_evars ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
y ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} apply HnotinFree.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
y ≠ fresh3
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} intros Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
Contra: y = fresh3
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} symmetry in Contra.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
Contra: fresh3 = y
False
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} contradiction.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (ex , (ex , ϕ))
Hfry: evar_is_fresh_in y (ex , (ex , ϕ))
e: M
x': evar
Heqx': x' = fresh_evar ϕ^{evar:S dbi↦x}
y': evar
Heqy': y' = fresh_evar ϕ^{evar:S dbi↦y}
fresh3: evar
Hneqfr1: fresh3 ≠ x
Hneqfr2: fresh3 ≠ y
Hneqfr11: fresh3 ≠ x'
Hneqfr22: fresh3 ≠ y'
HnotinFree1: fresh3 ∉ free_evars ϕ^{evar:S dbi↦x}
HnotinFree2: fresh3 ∉ free_evars ϕ^{evar:S dbi↦y}
HnotinFree: fresh3 ∉ free_evars ϕ
Heqfresh3: fresh3 =
evar_fresh
  (elements
     ({[x]}
      ∪ ({[y]}
         ∪ ({[x']}
            ∪ ({[y']}
               ∪ (free_evars
                    ϕ^{evar:
                    S dbi↦x}
                  ∪ (free_evars
                     ϕ^{evar:
                     S dbi↦y}
                     ∪ 
                     free_evars ϕ)))))))
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y} =
eval
  (update_evar_val y c (update_evar_val fresh3 e ρ))
  ϕ^{evar:0↦fresh3}^{evar:dbi↦y}
          reflexivity.
        +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
eval (update_evar_val x c ρ) (mu , ϕ)^{evar:dbi↦x} =
eval (update_evar_val y c ρ) (mu , ϕ)^{evar:dbi↦y} rewrite 2!eval_mu_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
(let X :=
   fresh_svar
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ) in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X S0 (update_evar_val x c ρ)
        in
      eval ρ'
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar x) dbi ϕ)^{svar:0↦X})) =
(let X :=
   fresh_svar
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ) in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X S0 (update_evar_val y c ρ)
        in
      eval ρ'
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar y) dbi ϕ)^{svar:0↦X})) fold evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
(let X :=
   fresh_svar
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ) in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X S0 (update_evar_val x c ρ)
        in
      eval ρ'
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar x) dbi ϕ)^{svar:0↦X})) =
(let X :=
   fresh_svar
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ) in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X S0 (update_evar_val y c ρ)
        in
      eval ρ'
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar y) dbi ϕ)^{svar:0↦X})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val
          (fresh_svar
             ((fix bevar_subst
                 (psi : Pattern) (x : db_index)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x' =>
                     patt_free_evar x'
                 | patt_free_svar x' =>
                     patt_free_svar x'
                 | patt_bound_evar n =>
                     match compare_nat n x with
                     | Nat_less _ _ _ =>
                         patt_bound_evar n
                     | Nat_equal _ _ _ => psi
                     | Nat_greater _ _ _ =>
                         patt_bound_evar (Nat.pred n)
                     end
                 | patt_bound_svar n =>
                     patt_bound_svar n
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     bevar_subst psi x phi1 $
                     bevar_subst psi x phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     bevar_subst psi x phi1 --->
                     bevar_subst psi x phi2
                 | ex , phi' =>
                     ex , bevar_subst psi (S x) phi'
                 | mu , phi' =>
                     mu , bevar_subst psi x phi'
                 end) (patt_free_evar x) dbi ϕ)) S0
          (update_evar_val x c ρ))
       ((fix bevar_subst
           (psi : Pattern) (x : db_index)
           (phi : Pattern) {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x' => patt_free_evar x'
           | patt_free_svar x' => patt_free_svar x'
           | patt_bound_evar n =>
               match compare_nat n x with
               | Nat_less _ _ _ => patt_bound_evar n
               | Nat_equal _ _ _ => psi
               | Nat_greater _ _ _ =>
                   patt_bound_evar (Nat.pred n)
               end
           | patt_bound_svar n => patt_bound_svar n
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               bevar_subst psi x phi1 $
               bevar_subst psi x phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               bevar_subst psi x phi1 --->
               bevar_subst psi x phi2
           | ex , phi' =>
               ex , bevar_subst psi (S x) phi'
           | mu , phi' => mu , bevar_subst psi x phi'
           end) (patt_free_evar x) dbi ϕ)^{svar:0↦
       fresh_svar
         ((fix bevar_subst
             (psi : Pattern) (x : db_index)
             (phi : Pattern) {struct phi} : Pattern :=
             match phi with
             | patt_free_evar x' => patt_free_evar x'
             | patt_free_svar x' => patt_free_svar x'
             | patt_bound_evar n =>
                 match compare_nat n x with
                 | Nat_less _ _ _ => patt_bound_evar n
                 | Nat_equal _ _ _ => psi
                 | Nat_greater _ _ _ =>
                     patt_bound_evar (Nat.pred n)
                 end
             | patt_bound_svar n => patt_bound_svar n
             | patt_sym sigma => patt_sym sigma
             | phi1 $ phi2 =>
                 bevar_subst psi x phi1 $
                 bevar_subst psi x phi2
             | ⊥ => ⊥
             | phi1 ---> phi2 =>
                 bevar_subst psi x phi1 --->
                 bevar_subst psi x phi2
             | ex , phi' =>
                 ex , bevar_subst psi (S x) phi'
             | mu , phi' =>
                 mu , bevar_subst psi x phi'
             end) (patt_free_evar x) dbi ϕ)}) =
LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val
          (fresh_svar
             ((fix bevar_subst
                 (psi : Pattern) (x : db_index)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x' =>
                     patt_free_evar x'
                 | patt_free_svar x' =>
                     patt_free_svar x'
                 | patt_bound_evar n =>
                     match compare_nat n x with
                     | Nat_less _ _ _ =>
                         patt_bound_evar n
                     | Nat_equal _ _ _ => psi
                     | Nat_greater _ _ _ =>
                         patt_bound_evar (Nat.pred n)
                     end
                 | patt_bound_svar n =>
                     patt_bound_svar n
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     bevar_subst psi x phi1 $
                     bevar_subst psi x phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     bevar_subst psi x phi1 --->
                     bevar_subst psi x phi2
                 | ex , phi' =>
                     ex , bevar_subst psi (S x) phi'
                 | mu , phi' =>
                     mu , bevar_subst psi x phi'
                 end) (patt_free_evar y) dbi ϕ)) S0
          (update_evar_val y c ρ))
       ((fix bevar_subst
           (psi : Pattern) (x : db_index)
           (phi : Pattern) {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x' => patt_free_evar x'
           | patt_free_svar x' => patt_free_svar x'
           | patt_bound_evar n =>
               match compare_nat n x with
               | Nat_less _ _ _ => patt_bound_evar n
               | Nat_equal _ _ _ => psi
               | Nat_greater _ _ _ =>
                   patt_bound_evar (Nat.pred n)
               end
           | patt_bound_svar n => patt_bound_svar n
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               bevar_subst psi x phi1 $
               bevar_subst psi x phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               bevar_subst psi x phi1 --->
               bevar_subst psi x phi2
           | ex , phi' =>
               ex , bevar_subst psi (S x) phi'
           | mu , phi' => mu , bevar_subst psi x phi'
           end) (patt_free_evar y) dbi ϕ)^{svar:0↦
       fresh_svar
         ((fix bevar_subst
             (psi : Pattern) (x : db_index)
             (phi : Pattern) {struct phi} : Pattern :=
             match phi with
             | patt_free_evar x' => patt_free_evar x'
             | patt_free_svar x' => patt_free_svar x'
             | patt_bound_evar n =>
                 match compare_nat n x with
                 | Nat_less _ _ _ => patt_bound_evar n
                 | Nat_equal _ _ _ => psi
                 | Nat_greater _ _ _ =>
                     patt_bound_evar (Nat.pred n)
                 end
             | patt_bound_svar n => patt_bound_svar n
             | patt_sym sigma => patt_sym sigma
             | phi1 $ phi2 =>
                 bevar_subst psi x phi1 $
                 bevar_subst psi x phi2
             | ⊥ => ⊥
             | phi1 ---> phi2 =>
                 bevar_subst psi x phi1 --->
                 bevar_subst psi x phi2
             | ex , phi' =>
                 ex , bevar_subst psi (S x) phi'
             | mu , phi' =>
                 mu , bevar_subst psi x phi'
             end) (patt_free_evar y) dbi ϕ)})
          apply f_equal.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
(λ S0 : propset M,
   eval
     (update_svar_val
        (fresh_svar
           ((fix bevar_subst
               (psi : Pattern) (x : db_index)
               (phi : Pattern) {struct phi} :
                 Pattern :=
               match phi with
               | patt_free_evar x' =>
                   patt_free_evar x'
               | patt_free_svar x' =>
                   patt_free_svar x'
               | patt_bound_evar n =>
                   match compare_nat n x with
                   | Nat_less _ _ _ =>
                       patt_bound_evar n
                   | Nat_equal _ _ _ => psi
                   | Nat_greater _ _ _ =>
                       patt_bound_evar (Nat.pred n)
                   end
               | patt_bound_svar n =>
                   patt_bound_svar n
               | patt_sym sigma => patt_sym sigma
               | phi1 $ phi2 =>
                   bevar_subst psi x phi1 $
                   bevar_subst psi x phi2
               | ⊥ => ⊥
               | phi1 ---> phi2 =>
                   bevar_subst psi x phi1 --->
                   bevar_subst psi x phi2
               | ex , phi' =>
                   ex , bevar_subst psi (S x) phi'
               | mu , phi' =>
                   mu , bevar_subst psi x phi'
               end) (patt_free_evar x) dbi ϕ)) S0
        (update_evar_val x c ρ))
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar x) dbi ϕ)^{svar:0↦fresh_svar
                                                 ((fix
                                                 bevar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_bound_svar
                                                 n =>
                                                 patt_bound_svar
                                                 n
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 $
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 --->
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bevar_subst
                                                 psi
                                                 (S x)
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bevar_subst
                                                 psi x
                                                 phi'
                                                 end)
                                                 (patt_free_evar
                                                 x)
                                                 dbi ϕ)}) =
(λ S0 : propset M,
   eval
     (update_svar_val
        (fresh_svar
           ((fix bevar_subst
               (psi : Pattern) (x : db_index)
               (phi : Pattern) {struct phi} :
                 Pattern :=
               match phi with
               | patt_free_evar x' =>
                   patt_free_evar x'
               | patt_free_svar x' =>
                   patt_free_svar x'
               | patt_bound_evar n =>
                   match compare_nat n x with
                   | Nat_less _ _ _ =>
                       patt_bound_evar n
                   | Nat_equal _ _ _ => psi
                   | Nat_greater _ _ _ =>
                       patt_bound_evar (Nat.pred n)
                   end
               | patt_bound_svar n =>
                   patt_bound_svar n
               | patt_sym sigma => patt_sym sigma
               | phi1 $ phi2 =>
                   bevar_subst psi x phi1 $
                   bevar_subst psi x phi2
               | ⊥ => ⊥
               | phi1 ---> phi2 =>
                   bevar_subst psi x phi1 --->
                   bevar_subst psi x phi2
               | ex , phi' =>
                   ex , bevar_subst psi (S x) phi'
               | mu , phi' =>
                   mu , bevar_subst psi x phi'
               end) (patt_free_evar y) dbi ϕ)) S0
        (update_evar_val y c ρ))
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             bevar_subst psi x phi1 $
             bevar_subst psi x phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             bevar_subst psi x phi1 --->
             bevar_subst psi x phi2
         | ex , phi' =>
             ex , bevar_subst psi (S x) phi'
         | mu , phi' => mu , bevar_subst psi x phi'
         end) (patt_free_evar y) dbi ϕ)^{svar:0↦fresh_svar
                                                 ((fix
                                                 bevar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_bound_svar
                                                 n =>
                                                 patt_bound_svar
                                                 n
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 $
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 --->
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bevar_subst
                                                 psi
                                                 (S x)
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bevar_subst
                                                 psi x
                                                 phi'
                                                 end)
                                                 (patt_free_evar
                                                 y)
                                                 dbi ϕ)}) apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
∀ x0 : propset M,
  eval
    (update_svar_val
       (fresh_svar
          ((fix bevar_subst
              (psi : Pattern) (x : db_index)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x' => patt_free_evar x'
              | patt_free_svar x' => patt_free_svar x'
              | patt_bound_evar n =>
                  match compare_nat n x with
                  | Nat_less _ _ _ =>
                      patt_bound_evar n
                  | Nat_equal _ _ _ => psi
                  | Nat_greater _ _ _ =>
                      patt_bound_evar (Nat.pred n)
                  end
              | patt_bound_svar n => patt_bound_svar n
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  bevar_subst psi x phi1 $
                  bevar_subst psi x phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  bevar_subst psi x phi1 --->
                  bevar_subst psi x phi2
              | ex , phi' =>
                  ex , bevar_subst psi (S x) phi'
              | mu , phi' =>
                  mu , bevar_subst psi x phi'
              end) (patt_free_evar x) dbi ϕ)) x0
       (update_evar_val x c ρ))
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar x) dbi ϕ)^{svar:0↦fresh_svar
                                                 ((fix
                                                 bevar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_bound_svar
                                                 n =>
                                                 patt_bound_svar
                                                 n
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 $
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 --->
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bevar_subst
                                                 psi
                                                 (S x)
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bevar_subst
                                                 psi x
                                                 phi'
                                                 end)
                                                 (patt_free_evar
                                                 x)
                                                 dbi ϕ)} =
  eval
    (update_svar_val
       (fresh_svar
          ((fix bevar_subst
              (psi : Pattern) (x : db_index)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x' => patt_free_evar x'
              | patt_free_svar x' => patt_free_svar x'
              | patt_bound_evar n =>
                  match compare_nat n x with
                  | Nat_less _ _ _ =>
                      patt_bound_evar n
                  | Nat_equal _ _ _ => psi
                  | Nat_greater _ _ _ =>
                      patt_bound_evar (Nat.pred n)
                  end
              | patt_bound_svar n => patt_bound_svar n
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  bevar_subst psi x phi1 $
                  bevar_subst psi x phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  bevar_subst psi x phi1 --->
                  bevar_subst psi x phi2
              | ex , phi' =>
                  ex , bevar_subst psi (S x) phi'
              | mu , phi' =>
                  mu , bevar_subst psi x phi'
              end) (patt_free_evar y) dbi ϕ)) x0
       (update_evar_val y c ρ))
    ((fix bevar_subst
        (psi : Pattern) (x : db_index) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x' => patt_free_evar x'
        | patt_free_svar x' => patt_free_svar x'
        | patt_bound_evar n =>
            match compare_nat n x with
            | Nat_less _ _ _ => patt_bound_evar n
            | Nat_equal _ _ _ => psi
            | Nat_greater _ _ _ =>
                patt_bound_evar (Nat.pred n)
            end
        | patt_bound_svar n => patt_bound_svar n
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            bevar_subst psi x phi1 $
            bevar_subst psi x phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            bevar_subst psi x phi1 --->
            bevar_subst psi x phi2
        | ex , phi' => ex , bevar_subst psi (S x) phi'
        | mu , phi' => mu , bevar_subst psi x phi'
        end) (patt_free_evar y) dbi ϕ)^{svar:0↦fresh_svar
                                                 ((fix
                                                 bevar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_bound_svar
                                                 n =>
                                                 patt_bound_svar
                                                 n
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 $
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 --->
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bevar_subst
                                                 psi
                                                 (S x)
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bevar_subst
                                                 psi x
                                                 phi'
                                                 end)
                                                 (patt_free_evar
                                                 y)
                                                 dbi ϕ)}
          intros S'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val
     (fresh_svar
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar x) dbi ϕ)) S'
     (update_evar_val x c ρ))
  ((fix bevar_subst
      (psi : Pattern) (x : db_index) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          bevar_subst psi x phi1 $
          bevar_subst psi x phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          bevar_subst psi x phi1 --->
          bevar_subst psi x phi2
      | ex , phi' => ex , bevar_subst psi (S x) phi'
      | mu , phi' => mu , bevar_subst psi x phi'
      end) (patt_free_evar x) dbi ϕ)^{svar:0↦fresh_svar
                                               ((fix
                                                 bevar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_bound_svar
                                                 n =>
                                                 patt_bound_svar
                                                 n
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 $
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 --->
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bevar_subst
                                                 psi
                                                 (S x)
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bevar_subst
                                                 psi x
                                                 phi'
                                                 end)
                                                 (patt_free_evar
                                                 x)
                                                 dbi ϕ)} =
eval
  (update_svar_val
     (fresh_svar
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar y) dbi ϕ)) S'
     (update_evar_val y c ρ))
  ((fix bevar_subst
      (psi : Pattern) (x : db_index) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          bevar_subst psi x phi1 $
          bevar_subst psi x phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          bevar_subst psi x phi1 --->
          bevar_subst psi x phi2
      | ex , phi' => ex , bevar_subst psi (S x) phi'
      | mu , phi' => mu , bevar_subst psi x phi'
      end) (patt_free_evar y) dbi ϕ)^{svar:0↦fresh_svar
                                               ((fix
                                                 bevar_subst
                                                 (psi : Pattern)
                                                 (x : db_index)
                                                 (phi : Pattern)
                                                 {struct
                                                 phi} :
                                                 Pattern :=
                                                 match
                                                 phi
                                                 with
                                                 | patt_free_evar
                                                 x' =>
                                                 patt_free_evar
                                                 x'
                                                 | patt_free_svar
                                                 x' =>
                                                 patt_free_svar
                                                 x'
                                                 | patt_bound_evar
                                                 n =>
                                                 match
                                                 compare_nat
                                                 n x
                                                 with
                                                 | Nat_less
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 n
                                                 | Nat_equal
                                                 _ _ _ =>
                                                 psi
                                                 | Nat_greater
                                                 _ _ _ =>
                                                 patt_bound_evar
                                                 (Nat.pred
                                                 n)
                                                 end
                                                 | patt_bound_svar
                                                 n =>
                                                 patt_bound_svar
                                                 n
                                                 | patt_sym
                                                 sigma =>
                                                 patt_sym
                                                 sigma
                                                 | phi1 $
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 $
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ⊥ =>
                                                 ⊥
                                                 | phi1 --->
                                                 phi2 =>
                                                 bevar_subst
                                                 psi x
                                                 phi1 --->
                                                 bevar_subst
                                                 psi x
                                                 phi2
                                                 | ex ,
                                                 phi' =>
                                                 ex ,
                                                 bevar_subst
                                                 psi
                                                 (S x)
                                                 phi'
                                                 | mu ,
                                                 phi' =>
                                                 mu ,
                                                 bevar_subst
                                                 psi x
                                                 phi'
                                                 end)
                                                 (patt_free_evar
                                                 y)
                                                 dbi ϕ)}
          rewrite -2!svar_open_evar_open_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val
     (fresh_svar
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar x) dbi ϕ)) S'
     (update_evar_val x c ρ))
  ϕ^{svar:0↦fresh_svar
              ((fix bevar_subst
                  (psi : Pattern) (x : db_index)
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x' =>
                      patt_free_evar x'
                  | patt_free_svar x' =>
                      patt_free_svar x'
                  | patt_bound_evar n =>
                      match compare_nat n x with
                      | Nat_less _ _ _ =>
                          patt_bound_evar n
                      | Nat_equal _ _ _ => psi
                      | Nat_greater _ _ _ =>
                          patt_bound_evar (Nat.pred n)
                      end
                  | patt_bound_svar n =>
                      patt_bound_svar n
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                      bevar_subst psi x phi1 $
                      bevar_subst psi x phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                      bevar_subst psi x phi1 --->
                      bevar_subst psi x phi2
                  | ex , phi' =>
                      ex , bevar_subst psi (S x) phi'
                  | mu , phi' =>
                      mu , bevar_subst psi x phi'
                  end) (patt_free_evar x) dbi ϕ)}^{evar:dbi↦x} =
eval
  (update_svar_val
     (fresh_svar
        ((fix bevar_subst
            (psi : Pattern) (x : db_index)
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x' => patt_free_evar x'
            | patt_free_svar x' => patt_free_svar x'
            | patt_bound_evar n =>
                match compare_nat n x with
                | Nat_less _ _ _ => patt_bound_evar n
                | Nat_equal _ _ _ => psi
                | Nat_greater _ _ _ =>
                    patt_bound_evar (Nat.pred n)
                end
            | patt_bound_svar n => patt_bound_svar n
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                bevar_subst psi x phi1 $
                bevar_subst psi x phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                bevar_subst psi x phi1 --->
                bevar_subst psi x phi2
            | ex , phi' =>
                ex , bevar_subst psi (S x) phi'
            | mu , phi' => mu , bevar_subst psi x phi'
            end) (patt_free_evar y) dbi ϕ)) S'
     (update_evar_val y c ρ))
  ϕ^{svar:0↦fresh_svar
              ((fix bevar_subst
                  (psi : Pattern) (x : db_index)
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x' =>
                      patt_free_evar x'
                  | patt_free_svar x' =>
                      patt_free_svar x'
                  | patt_bound_evar n =>
                      match compare_nat n x with
                      | Nat_less _ _ _ =>
                          patt_bound_evar n
                      | Nat_equal _ _ _ => psi
                      | Nat_greater _ _ _ =>
                          patt_bound_evar (Nat.pred n)
                      end
                  | patt_bound_svar n =>
                      patt_bound_svar n
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                      bevar_subst psi x phi1 $
                      bevar_subst psi x phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                      bevar_subst psi x phi1 --->
                      bevar_subst psi x phi2
                  | ex , phi' =>
                      ex , bevar_subst psi (S x) phi'
                  | mu , phi' =>
                      mu , bevar_subst psi x phi'
                  end) (patt_free_evar y) dbi ϕ)}^{evar:dbi↦y}
          fold bevar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
     (update_evar_val x c ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦x} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y}
          rewrite -2!update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val x c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦x} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y}
          rewrite (proj2 (IHsz _ _ _ _) x y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
size
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in x
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y} rewrite -svar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in x
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in x
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y}
          apply evar_is_fresh_in_mu in Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , (mu , ϕ))
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in x
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y}
          apply evar_is_fresh_in_mu in Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , (mu , ϕ))
Hfry: evar_is_fresh_in y (mu , (mu , ϕ))
S': propset M
evar_is_fresh_in x
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y}
          apply evar_fresh_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , (mu , ϕ))
Hfry: evar_is_fresh_in y (mu , (mu , ϕ))
S': propset M
evar_is_fresh_in x ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y} apply Hfrx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y}
          apply evar_fresh_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
evar_is_fresh_in y ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y} apply Hfry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar x]}^{evar:dbi↦y} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{svar:0↦fresh_svar ϕ^[evar:dbi↦patt_free_evar y]}^{evar:dbi↦y}
          rewrite 2!svar_open_evar_open_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
        ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦fresh_svar
                           ϕ^[evar:dbi↦patt_free_evar
                                         x]} =
eval
  (update_evar_val y c
     (update_svar_val
        (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
        ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦fresh_svar
                           ϕ^[evar:dbi↦patt_free_evar
                                         y]}
          rewrite 2!update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦fresh_svar
                           ϕ^[evar:dbi↦patt_free_evar
                                         x]} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦fresh_svar
                           ϕ^[evar:dbi↦patt_free_evar
                                         y]}
          rewrite (proj1 (IHsz _ _ _ _) _ (fresh_svar (ϕ^{evar: dbi ↦ y}))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
size ϕ^{evar:dbi↦y} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in
  (fresh_svar ϕ^[evar:dbi↦patt_free_evar x])
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in (fresh_svar ϕ^{evar:dbi↦y})
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val (fresh_svar ϕ^{evar:dbi↦y}) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^{evar:dbi↦y}} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^[evar:dbi↦
    patt_free_evar y]}
          rewrite -evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
size ϕ ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in
  (fresh_svar ϕ^[evar:dbi↦patt_free_evar x])
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in (fresh_svar ϕ^{evar:dbi↦y})
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val (fresh_svar ϕ^{evar:dbi↦y}) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^{evar:dbi↦y}} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^[evar:dbi↦
    patt_free_evar y]} lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in
  (fresh_svar ϕ^[evar:dbi↦patt_free_evar x])
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in (fresh_svar ϕ^{evar:dbi↦y})
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val (fresh_svar ϕ^{evar:dbi↦y}) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^{evar:dbi↦y}} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^[evar:dbi↦
    patt_free_evar y]}
          apply svar_fresh_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in
  (fresh_svar ϕ^[evar:dbi↦patt_free_evar x]) ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in (fresh_svar ϕ^{evar:dbi↦y})
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val (fresh_svar ϕ^{evar:dbi↦y}) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^{evar:dbi↦y}} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^[evar:dbi↦
    patt_free_evar y]}
          rewrite fresh_svar_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in (fresh_svar ϕ) ϕ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in (fresh_svar ϕ^{evar:dbi↦y})
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val (fresh_svar ϕ^{evar:dbi↦y}) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^{evar:dbi↦y}} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^[evar:dbi↦
    patt_free_evar y]} apply set_svar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
svar_is_fresh_in (fresh_svar ϕ^{evar:dbi↦y})
  ϕ^{evar:dbi↦y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val (fresh_svar ϕ^{evar:dbi↦y}) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^{evar:dbi↦y}} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦
  fresh_svar ϕ^[evar:dbi↦
    patt_free_evar y]}
          apply set_svar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (ϕ : Pattern) (dbi : db_index) 
  (ρ : Valuation),
  size ϕ ≤ sz
  → (∀ (X Y : svar) (S : propset M),
       svar_is_fresh_in X ϕ
       → svar_is_fresh_in Y ϕ
         → eval (update_svar_val X S ρ)
             ϕ^{svar:dbi↦X} =
           eval (update_svar_val Y S ρ)
             ϕ^{svar:dbi↦Y})
    ∧ (∀ (x y : evar) (c : M),
         evar_is_fresh_in x ϕ
         → evar_is_fresh_in y ϕ
           → eval (update_evar_val x c ρ)
               ϕ^{evar:dbi↦x} =
             eval (update_evar_val y c ρ)
               ϕ^{evar:dbi↦y})
ϕ: Pattern
dbi: db_index
ρ: Valuation
Hsz: S (size ϕ) ≤ S sz
x, y: evar
c: M
Hfrx: evar_is_fresh_in x (mu , ϕ)
Hfry: evar_is_fresh_in y (mu , ϕ)
S': propset M
eval
  (update_svar_val (fresh_svar ϕ^{evar:dbi↦y}) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦fresh_svar ϕ^{evar:dbi↦y}} =
eval
  (update_svar_val
     (fresh_svar ϕ^[evar:dbi↦patt_free_evar y]) S'
     (update_evar_val y c ρ))
  ϕ^{evar:dbi↦y}^{svar:0↦fresh_svar
                           ϕ^[evar:dbi↦patt_free_evar
                                         y]}
          reflexivity.
      }
  Qed.


  Lemma eval_fresh_evar_open ϕ x y c dbi ρ:
    evar_is_fresh_in x ϕ ->
    evar_is_fresh_in y ϕ ->
    @eval M (update_evar_val x c ρ) (ϕ^{evar: dbi ↦ x})
    = eval (update_evar_val y c ρ) (ϕ^{evar: dbi ↦ y}).
signature: Signature
M: Model
ϕ: Pattern
x, y: evar
c: M
dbi: db_index
ρ: Valuation
evar_is_fresh_in x ϕ
→ evar_is_fresh_in y ϕ
  → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
    eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
  Proof.
signature: Signature
M: Model
ϕ: Pattern
x, y: evar
c: M
dbi: db_index
ρ: Valuation
evar_is_fresh_in x ϕ
→ evar_is_fresh_in y ϕ
  → eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
    eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
    move=> Hfrx Hfry.
signature: Signature
M: Model
ϕ: Pattern
x, y: evar
c: M
dbi: db_index
ρ: Valuation
Hfrx: evar_is_fresh_in x ϕ
Hfry: evar_is_fresh_in y ϕ
eval (update_evar_val x c ρ) ϕ^{evar:dbi↦x} =
eval (update_evar_val y c ρ) ϕ^{evar:dbi↦y}
    eapply (proj2 (Private_eval_fresh_var_open (size ϕ) ϕ dbi ρ _) x y c); assumption.
    Unshelve.
signature: Signature
M: Model
ϕ: Pattern
x, y: evar
c: M
dbi: db_index
ρ: Valuation
Hfrx: evar_is_fresh_in x ϕ
Hfry: evar_is_fresh_in y ϕ
size ϕ ≤ size ϕ lia.
  Qed.

  Lemma eval_fresh_svar_open ϕ X Y S dbi ρ:
    svar_is_fresh_in X ϕ ->
    svar_is_fresh_in Y ϕ ->
    @eval M (update_svar_val X S ρ) (ϕ^{svar: dbi ↦ X})
    = eval (update_svar_val Y S ρ) (ϕ^{svar: dbi ↦ Y}).
signature: Signature
M: Model
ϕ: Pattern
X, Y: svar
S: propset M
dbi: db_index
ρ: Valuation
svar_is_fresh_in X ϕ
→ svar_is_fresh_in Y ϕ
  → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
    eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
  Proof.
signature: Signature
M: Model
ϕ: Pattern
X, Y: svar
S: propset M
dbi: db_index
ρ: Valuation
svar_is_fresh_in X ϕ
→ svar_is_fresh_in Y ϕ
  → eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
    eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
    move=> HfrX HfrY.
signature: Signature
M: Model
ϕ: Pattern
X, Y: svar
S: propset M
dbi: db_index
ρ: Valuation
HfrX: svar_is_fresh_in X ϕ
HfrY: svar_is_fresh_in Y ϕ
eval (update_svar_val X S ρ) ϕ^{svar:dbi↦X} =
eval (update_svar_val Y S ρ) ϕ^{svar:dbi↦Y}
    eapply (proj1 (Private_eval_fresh_var_open (size ϕ) ϕ dbi ρ _) X Y S); assumption.
    Unshelve.
signature: Signature
M: Model
ϕ: Pattern
X, Y: svar
S: propset M
dbi: db_index
ρ: Valuation
HfrX: svar_is_fresh_in X ϕ
HfrY: svar_is_fresh_in Y ϕ
size ϕ ≤ size ϕ lia.
  Qed.

  (* There are two ways how to plug a pattern phi2 into a pattern phi1:
   either substitute it for some variable,
   or evaluate phi2 first and then evaluate phi1 with valuation updated to the result of phi2
  *)
  Lemma Private_plugging_patterns : forall (sz : nat) (dbi : db_index) (phi1 phi2 : Pattern),
      size phi1 <= sz -> forall (ρ : Valuation) (X : svar),
        well_formed_closed phi2 -> well_formed_closed_mu_aux phi1 (S dbi) ->
        ~ elem_of X (free_svars phi1) ->
        @eval M ρ (phi1^[svar:dbi ↦ phi2])
        = eval (update_svar_val X (@eval M ρ phi2) ρ)
                                (phi1^{svar: dbi ↦ X}).
signature: Signature
M: Model
∀ (sz : nat) (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval (update_svar_val X (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
  Proof.
signature: Signature
M: Model
∀ (sz : nat) (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval (update_svar_val X (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
    induction sz; intros dbi phi1 phi2 Hsz ρ X Hwfc2 Hwfphi1 H.
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
Hsz: size phi1 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 (S dbi)
H: X ∉ free_svars phi1
eval ρ phi1^[svar:dbi↦phi2] =
eval (update_svar_val X (eval ρ phi2) ρ)
  phi1^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size phi1 ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 (S dbi)
H: X ∉ free_svars phi1
eval ρ phi1^[svar:dbi↦phi2] =
eval (update_svar_val X (eval ρ phi2) ρ)
  phi1^{svar:dbi↦X}
    - (* sz == 0 *)
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
Hsz: size phi1 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 (S dbi)
H: X ∉ free_svars phi1
eval ρ phi1^[svar:dbi↦phi2] =
eval (update_svar_val X (eval ρ phi2) ρ)
  phi1^{svar:dbi↦X}
      destruct phi1; simpl in Hsz; simpl.
signature: Signature
M: Model
dbi: db_index
x: evar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_evar x) (S dbi)
H: X ∉ free_svars (patt_free_evar x)
eval ρ (patt_free_evar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_evar x)^{svar:dbi↦X}
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)^{svar:dbi↦X}
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_evar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_evar n)
eval ρ (patt_bound_evar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_evar n)^{svar:dbi↦X}
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)^{svar:dbi↦X}
signature: Signature
M: Model
dbi: db_index
sigma: symbols
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_sym sigma) (S dbi)
H: X ∉ free_svars (patt_sym sigma)
eval ρ (patt_sym sigma) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_sym sigma)^{svar:dbi↦X}
signature: Signature
M: Model
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars (phi1_1 $ phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 $ phi1_2)^{svar:dbi↦X}
signature: Signature
M: Model
dbi: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux ⊥ (S dbi)
H: X ∉ free_svars ⊥
eval ρ ⊥ =
eval (update_svar_val X (eval ρ phi2) ρ)
  ⊥^{svar:dbi↦X}
signature: Signature
M: Model
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars (phi1_1 ---> phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 ---> phi1_2)^{svar:dbi↦X}
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars (ex , phi1)
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1)^{svar:dbi↦X}
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
eval ρ (mu , phi1^[svar:S dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (mu , phi1)^{svar:dbi↦X}
      + (* free_evar *)
signature: Signature
M: Model
dbi: db_index
x: evar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_evar x) (S dbi)
H: X ∉ free_svars (patt_free_evar x)
eval ρ (patt_free_evar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_evar x)^{svar:dbi↦X}
        repeat rewrite -> eval_free_evar_simpl.
signature: Signature
M: Model
dbi: db_index
x: evar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_evar x) (S dbi)
H: X ∉ free_svars (patt_free_evar x)
{[evar_valuation ρ x]} =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_evar x)^{svar:dbi↦X} auto.
      + (* free_svar *)
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)^{svar:dbi↦X}
        unfold svar_open.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)^[svar:dbi↦patt_free_svar X] simpl.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)
        do 2 rewrite -> eval_free_svar_simpl.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
svar_valuation ρ x =
svar_valuation (update_svar_val X (eval ρ phi2) ρ) x
        unfold update_svar_val.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
svar_valuation ρ x =
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then eval ρ phi2
        else svar_valuation ρ sv'
  |} x
        destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
svar_valuation
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} x =
svar_valuation
  {|
    evar_valuation :=
      evar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |};
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then
         eval
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} phi2
        else
         svar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} sv'
  |} x simpl.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
ρₛ x =
(if decide (X = x)
 then
  eval
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
    phi2
 else ρₛ x)
        destruct (decide (X = x)).
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
n: X ≠ x
ρₛ x = ρₛ x
        *
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 simpl in H.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ {[x]}
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 simpl.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ {[x]}
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 unfold not in H.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 exfalso.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
False apply H.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
X ∈ {[x]}
          apply elem_of_singleton_2.
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
X = x auto.
        *
signature: Signature
M: Model
dbi: db_index
x: svar
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
n: X ≠ x
ρₛ x = ρₛ x auto.
      + (* bound_evar *)
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_evar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_evar n)
eval ρ (patt_bound_evar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_evar n)^{svar:dbi↦X}
        rewrite 2!eval_bound_evar_simpl.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_evar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_evar n)
∅ = ∅
        reflexivity.
      + (* bound_svar *)
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)^{svar:dbi↦X}
        unfold svar_open.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)^[svar:dbi↦patt_free_svar X] simpl.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end  case_match; simpl.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_svar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar X)
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_svar (Nat.pred n)) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar (Nat.pred n))
        *
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_svar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n) repeat rewrite -> eval_bound_svar_simpl; auto.
        *
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar X) rewrite eval_free_svar_simpl.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
svar_valuation (update_svar_val X (eval ρ phi2) ρ) X unfold update_svar_val.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then eval ρ phi2
        else svar_valuation ρ sv'
  |} X
          destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 =
svar_valuation
  {|
    evar_valuation :=
      evar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |};
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then
         eval
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} phi2
        else
         svar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} sv'
  |} X simpl.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 =
(if decide (X = X)
 then
  eval
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
    phi2
 else ρₛ X)
          destruct (decide (X = X)).
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
e0: X = X
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
n0: X ≠ X
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 = ρₛ X auto.
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
n0: X ≠ X
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 = ρₛ X contradiction.
        *
signature: Signature
M: Model
dbi, n: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_svar (Nat.pred n)) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar (Nat.pred n)) repeat rewrite -> eval_bound_svar_simpl; auto.
      + (* sym *)
signature: Signature
M: Model
dbi: db_index
sigma: symbols
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_sym sigma) (S dbi)
H: X ∉ free_svars (patt_sym sigma)
eval ρ (patt_sym sigma) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_sym sigma)^{svar:dbi↦X}
        simpl.
signature: Signature
M: Model
dbi: db_index
sigma: symbols
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_sym sigma) (S dbi)
H: X ∉ free_svars (patt_sym sigma)
eval ρ (patt_sym sigma) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_sym sigma)^{svar:dbi↦X} repeat rewrite -> eval_sym_simpl; auto.
      + (* app *)
signature: Signature
M: Model
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars (phi1_1 $ phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 $ phi1_2)^{svar:dbi↦X}
        lia.
      + (* bot *)
signature: Signature
M: Model
dbi: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux ⊥ (S dbi)
H: X ∉ free_svars ⊥
eval ρ ⊥ =
eval (update_svar_val X (eval ρ phi2) ρ)
  ⊥^{svar:dbi↦X}
        simpl.
signature: Signature
M: Model
dbi: db_index
phi2: Pattern
Hsz: 0 ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux ⊥ (S dbi)
H: X ∉ free_svars ⊥
eval ρ ⊥ =
eval (update_svar_val X (eval ρ phi2) ρ)
  ⊥^{svar:dbi↦X} repeat rewrite eval_bott_simpl; auto.
      + (* impl *)
signature: Signature
M: Model
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars (phi1_1 ---> phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 ---> phi1_2)^{svar:dbi↦X}
        lia.
      + (* ex *)
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars (ex , phi1)
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1)^{svar:dbi↦X}
        lia.
      + (* mu *)
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ 0
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
eval ρ (mu , phi1^[svar:S dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (mu , phi1)^{svar:dbi↦X}
        lia.
    - (* sz = S sz' *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size phi1 ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 (S dbi)
H: X ∉ free_svars phi1
eval ρ phi1^[svar:dbi↦phi2] =
eval (update_svar_val X (eval ρ phi2) ρ)
  phi1^{svar:dbi↦X}
      destruct phi1; simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: evar
phi2: Pattern
Hsz: size (patt_free_evar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_evar x) (S dbi)
H: X ∉ free_svars (patt_free_evar x)
eval ρ (patt_free_evar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_evar x)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_evar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_evar n)
eval ρ (patt_bound_evar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_evar n)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
sigma: symbols
phi2: Pattern
Hsz: size (patt_sym sigma) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_sym sigma) (S dbi)
H: X ∉ free_svars (patt_sym sigma)
eval ρ (patt_sym sigma) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_sym sigma)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 $ phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars (phi1_1 $ phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 $ phi1_2)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi2: Pattern
Hsz: size ⊥ ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux ⊥ (S dbi)
H: X ∉ free_svars ⊥
eval ρ ⊥ =
eval (update_svar_val X (eval ρ phi2) ρ)
  ⊥^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 ---> phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars (phi1_1 ---> phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 ---> phi1_2)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (ex , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars (ex , phi1)
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1)^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
eval ρ (mu , phi1^[svar:S dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (mu , phi1)^{svar:dbi↦X}
      (* HERE we duplicate some of the effort. I do not like it. *)
      + (* free_evar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: evar
phi2: Pattern
Hsz: size (patt_free_evar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_evar x) (S dbi)
H: X ∉ free_svars (patt_free_evar x)
eval ρ (patt_free_evar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_evar x)^{svar:dbi↦X}
        repeat rewrite eval_free_evar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: evar
phi2: Pattern
Hsz: size (patt_free_evar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_evar x) (S dbi)
H: X ∉ free_svars (patt_free_evar x)
{[evar_valuation ρ x]} =
{[evar_valuation (update_svar_val X (eval ρ phi2) ρ) x]} auto.
      + (* free_svar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)^{svar:dbi↦X}
        unfold svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)^[svar:dbi↦patt_free_svar X] simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
eval ρ (patt_free_svar x) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar x)
        do 2 rewrite -> eval_free_svar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
svar_valuation ρ x =
svar_valuation (update_svar_val X (eval ρ phi2) ρ) x
        unfold update_svar_val.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
svar_valuation ρ x =
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then eval ρ phi2
        else svar_valuation ρ sv'
  |} x
        destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
svar_valuation
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} x =
svar_valuation
  {|
    evar_valuation :=
      evar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |};
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then
         eval
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} phi2
        else
         svar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} sv'
  |} x simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
ρₛ x =
(if decide (X = x)
 then
  eval
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
    phi2
 else ρₛ x)
        destruct (decide (X = x)).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
n: X ≠ x
ρₛ x = ρₛ x
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ {[x]}
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ {[x]}
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 unfold not in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
ρₛ x =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 exfalso.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
False apply H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
X ∈ {[x]}
          apply elem_of_singleton_2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∈ {[x]} → False
e: X = x
X = x auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
x: svar
phi2: Pattern
Hsz: size (patt_free_svar x) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_free_svar x) (S dbi)
H: X ∉ free_svars (patt_free_svar x)
n: X ≠ x
ρₛ x = ρₛ x auto.
      + (* bound_evar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_evar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_evar n)
eval ρ (patt_bound_evar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_evar n)^{svar:dbi↦X}
        rewrite !eval_bound_evar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_evar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_evar n)
∅ = ∅ reflexivity.
      + (* bound_svar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)^{svar:dbi↦X}
        unfold svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)^[svar:dbi↦patt_free_svar X] simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => phi2
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end =
eval (update_svar_val X (eval ρ phi2) ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_svar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar X)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_svar (Nat.pred n)) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar (Nat.pred n))
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_svar n) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar n) repeat rewrite -> eval_bound_evar_simpl; auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_free_svar X) rewrite eval_free_svar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
svar_valuation (update_svar_val X (eval ρ phi2) ρ) X unfold update_svar_val.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ phi2 =
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then eval ρ phi2
        else svar_valuation ρ sv'
  |} X
          destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 =
svar_valuation
  {|
    evar_valuation :=
      evar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |};
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then
         eval
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} phi2
        else
         svar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} sv'
  |} X simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 =
(if decide (X = X)
 then
  eval
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
    phi2
 else ρₛ X)
          rewrite decide_eq_same.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  phi2 reflexivity.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi, n: db_index
phi2: Pattern
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (patt_bound_svar n) 
  (S dbi)
H: X ∉ free_svars (patt_bound_svar n)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_svar (Nat.pred n)) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_bound_svar (Nat.pred n)) repeat rewrite -> eval_bound_svar_simpl; auto.
      + (* sym *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
sigma: symbols
phi2: Pattern
Hsz: size (patt_sym sigma) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_sym sigma) (S dbi)
H: X ∉ free_svars (patt_sym sigma)
eval ρ (patt_sym sigma) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_sym sigma)^{svar:dbi↦X}
        simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
sigma: symbols
phi2: Pattern
Hsz: size (patt_sym sigma) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (patt_sym sigma) (S dbi)
H: X ∉ free_svars (patt_sym sigma)
eval ρ (patt_sym sigma) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (patt_sym sigma)^{svar:dbi↦X} repeat rewrite -> eval_sym_simpl; auto.
      (* HERE the duplication ends *)
      + (* app *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 $ phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars (phi1_1 $ phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 $ phi1_2)^{svar:dbi↦X} mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 $ phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars (phi1_1 $ phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} $ phi1_2^{svar:dbi↦X})
        simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 $ phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1 ∪ free_svars phi1_2
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} $ phi1_2^{svar:dbi↦X}) apply not_elem_of_union in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 $ phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: (X ∉ free_svars phi1_1) ∧ X ∉ free_svars phi1_2
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} $ phi1_2^{svar:dbi↦X}) destruct H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 $ phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
eval ρ
  (phi1_1^[svar:dbi↦phi2] $ phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} $ phi1_2^{svar:dbi↦X})
        do 2 rewrite -> eval_app_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 $ phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
app_ext (eval ρ phi1_1^[svar:dbi↦phi2])
  (eval ρ phi1_2^[svar:dbi↦phi2]) =
app_ext
  (eval (update_svar_val X (eval ρ phi2) ρ)
     phi1_1^{svar:dbi↦X})
  (eval (update_svar_val X (eval ρ phi2) ρ)
     phi1_2^{svar:dbi↦X})
        simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
app_ext (eval ρ phi1_1^[svar:dbi↦phi2])
  (eval ρ phi1_2^[svar:dbi↦phi2]) =
app_ext
  (eval (update_svar_val X (eval ρ phi2) ρ)
     phi1_1^{svar:dbi↦X})
  (eval (update_svar_val X (eval ρ phi2) ρ)
     phi1_2^{svar:dbi↦X})
        rewrite <- (IHsz _ phi1_1), <- (IHsz _ phi1_2).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
app_ext (eval ρ phi1_1^[svar:dbi↦phi2])
  (eval ρ phi1_2^[svar:dbi↦phi2]) =
app_ext (eval ρ phi1_1^[svar:dbi↦phi2])
  (eval ρ phi1_2^[svar:dbi↦phi2])
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_2 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_1
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
app_ext (eval ρ phi1_1^[svar:dbi↦phi2])
  (eval ρ phi1_2^[svar:dbi↦phi2]) =
app_ext (eval ρ phi1_1^[svar:dbi↦phi2])
  (eval ρ phi1_2^[svar:dbi↦phi2]) reflexivity.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_2 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2 assumption.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_2 (S dbi) now apply andb_true_iff in Hwfphi1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_2 auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_1 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2 assumption.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_1 (S dbi) now apply andb_true_iff in Hwfphi1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (phi1_1 $ phi1_2) (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_1 auto.
      + (* Bot. Again, a duplication of the sz=0 case *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi2: Pattern
Hsz: size ⊥ ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux ⊥ (S dbi)
H: X ∉ free_svars ⊥
eval ρ ⊥ =
eval (update_svar_val X (eval ρ phi2) ρ)
  ⊥^{svar:dbi↦X}
        simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi2: Pattern
Hsz: size ⊥ ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux ⊥ (S dbi)
H: X ∉ free_svars ⊥
eval ρ ⊥ =
eval (update_svar_val X (eval ρ phi2) ρ)
  ⊥^{svar:dbi↦X} repeat rewrite eval_bott_simpl; auto.
      + (* imp *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 ---> phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars (phi1_1 ---> phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1 ---> phi1_2)^{svar:dbi↦X}
        mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: size (phi1_1 ---> phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars (phi1_1 ---> phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} ---> phi1_2^{svar:dbi↦X})
        simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars (phi1_1 ---> phi1_2)
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} ---> phi1_2^{svar:dbi↦X}) simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1 ∪ free_svars phi1_2
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} ---> phi1_2^{svar:dbi↦X}) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1 ∪ free_svars phi1_2
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} ---> phi1_2^{svar:dbi↦X})
        apply not_elem_of_union in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: (X ∉ free_svars phi1_1) ∧ X ∉ free_svars phi1_2
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} ---> phi1_2^{svar:dbi↦X}) destruct H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
eval ρ
  (phi1_1^[svar:dbi↦phi2] ---> phi1_2^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (phi1_1^{svar:dbi↦X} ---> phi1_2^{svar:dbi↦X})
        do 2 rewrite -> eval_imp_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
⊤ ∖ eval ρ phi1_1^[svar:dbi↦phi2]
∪ eval ρ phi1_2^[svar:dbi↦phi2] =
⊤
∖ eval (update_svar_val X (eval ρ phi2) ρ)
    phi1_1^{svar:dbi↦X}
∪ eval (update_svar_val X (eval ρ phi2) ρ)
    phi1_2^{svar:dbi↦X}
        rewrite <- IHsz, <- IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
⊤ ∖ eval ρ phi1_1^[svar:dbi↦phi2]
∪ eval ρ phi1_2^[svar:dbi↦phi2] =
⊤ ∖ eval ρ phi1_1^[svar:dbi↦phi2]
∪ eval ρ phi1_2^[svar:dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_2 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_1
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
⊤ ∖ eval ρ phi1_1^[svar:dbi↦phi2]
∪ eval ρ phi1_2^[svar:dbi↦phi2] =
⊤ ∖ eval ρ phi1_1^[svar:dbi↦phi2]
∪ eval ρ phi1_2^[svar:dbi↦phi2] reflexivity.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_2 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2 assumption.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_2 (S dbi) now apply andb_true_iff in Hwfphi1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_2 auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
size phi1_1 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed phi2 assumption.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
well_formed_closed_mu_aux phi1_1 (S dbi) now apply andb_true_iff in Hwfphi1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1_1, phi1_2, phi2: Pattern
Hsz: S (size phi1_1 + size phi1_2) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux
  (phi1_1 ---> phi1_2) 
  (S dbi)
H: X ∉ free_svars phi1_1
H0: X ∉ free_svars phi1_2
X ∉ free_svars phi1_1 auto.

      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (ex , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars (ex , phi1)
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1)^{svar:dbi↦X} simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars (ex , phi1)
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1)^{svar:dbi↦X} simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1)^{svar:dbi↦X} unfold svar_open in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1)^[svar:dbi↦patt_free_svar X] simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
eval ρ (ex , phi1^[svar:dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (ex , phi1^[svar:dbi↦patt_free_svar X])
        do 2 rewrite -> eval_ex_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
(let x := fresh_evar phi1^[svar:dbi↦phi2] in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ' phi1^[svar:dbi↦phi2]^{evar:0↦x})) =
(let x := fresh_evar phi1^[svar:dbi↦patt_free_svar X]
   in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e
          (update_svar_val X (eval ρ phi2) ρ) in
      eval ρ'
        phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦x})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar phi1^[svar:dbi↦phi2]) e ρ)
       phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                      phi1^[svar:dbi↦phi2]}) =
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar phi1^[svar:dbi↦patt_free_svar X])
          e (update_svar_val X (eval ρ phi2) ρ))
       phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦fresh_evar
                                                 phi1^[svar:dbi↦
                                                 patt_free_svar
                                                 X]})
        apply propset_fa_union_same.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
∀ c : M,
  eval
    (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2])
       c ρ)
    phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                   phi1^[svar:dbi↦phi2]} =
  eval
    (update_evar_val
       (fresh_evar phi1^[svar:dbi↦patt_free_svar X]) c
       (update_svar_val X (eval ρ phi2) ρ))
    phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦fresh_evar
                                               phi1^[svar:dbi↦
                                               patt_free_svar
                                                 X]} intros c.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval
  (update_evar_val
     (fresh_evar phi1^[svar:dbi↦patt_free_svar X]) c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦fresh_evar
                                             phi1^[svar:dbi↦
                                             patt_free_svar
                                               X]}

        (* x = fresh_evar phi1' *)
        (* y = evar_fresh (elements (free_evars phi1') U (free_evars phi2)) *)

        remember (phi1^[svar: dbi ↦ (patt_free_svar X)]) as phi1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval
  (update_evar_val (fresh_evar phi1') c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1'^{evar:0↦fresh_evar phi1'}
        remember (fresh_evar phi1') as Xfr2'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval
  (update_evar_val Xfr2' c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1'^{evar:0↦Xfr2'}
        remember (evar_fresh (elements (union (free_evars phi1') (free_evars phi2)))) as Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval
  (update_evar_val Xfr2' c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1'^{evar:0↦Xfr2'}
        remember (update_svar_val X (eval ρ phi2) ρ) as ρ2'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval (update_evar_val Xfr2' c ρ2')
  phi1'^{evar:0↦Xfr2'}
        pose proof (Hfresh_subst := eval_fresh_evar_open phi1' Xfr2' Xu c 0 ρ2').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval (update_evar_val Xfr2' c ρ2')
  phi1'^{evar:0↦Xfr2'}

        rewrite -> Hfresh_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
evar_is_fresh_in Xfr2' phi1'
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
evar_is_fresh_in Xu phi1'
        2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
evar_is_fresh_in Xfr2' phi1' subst Xfr2'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in (fresh_evar phi1')
  phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val
         (fresh_evar phi1') c ρ2')
      phi1'^{evar:0↦fresh_evar phi1'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
evar_is_fresh_in (fresh_evar phi1') phi1' apply set_evar_fresh_is_fresh. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
evar_is_fresh_in Xu phi1'
        2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
evar_is_fresh_in Xu phi1' pose proof (Hfr := set_evar_fresh_is_fresh').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
Hfr: ∀ S : EVarSet, evar_fresh (elements S) ∉ S
evar_is_fresh_in Xu phi1'
             specialize (Hfr (union (free_evars phi1') (free_evars phi2))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
Hfr: evar_fresh
  (elements (free_evars phi1' ∪ free_evars phi2))
∉ free_evars phi1' ∪ free_evars phi2
evar_is_fresh_in Xu phi1'
             subst Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars phi1'
           ∪ free_evars phi2))) phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars phi1'
                ∪ free_evars phi2))) c
         ρ2')
      phi1'^{evar:0↦evar_fresh
                     (elements
                     (free_evars phi1'
                     ∪ 
                     free_evars phi2))}
Hfr: evar_fresh
  (elements (free_evars phi1' ∪ free_evars phi2))
∉ free_evars phi1' ∪ free_evars phi2
evar_is_fresh_in
  (evar_fresh
     (elements (free_evars phi1' ∪ free_evars phi2)))
  phi1'
             apply not_elem_of_union in Hfr.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars phi1'
           ∪ free_evars phi2))) phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars phi1'
                ∪ free_evars phi2))) c
         ρ2')
      phi1'^{evar:0↦evar_fresh
                     (elements
                     (free_evars phi1'
                     ∪ 
                     free_evars phi2))}
Hfr: (evar_fresh
   (elements
      (free_evars phi1' ∪ free_evars phi2))
 ∉ free_evars phi1')
∧ evar_fresh
    (elements
       (free_evars phi1' ∪ free_evars phi2))
  ∉ free_evars phi2
evar_is_fresh_in
  (evar_fresh
     (elements (free_evars phi1' ∪ free_evars phi2)))
  phi1' destruct Hfr as [Hfr _].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars phi1'
           ∪ free_evars phi2))) phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars phi1'
                ∪ free_evars phi2))) c
         ρ2')
      phi1'^{evar:0↦evar_fresh
                     (elements
                     (free_evars phi1'
                     ∪ 
                     free_evars phi2))}
Hfr: evar_fresh
  (elements (free_evars phi1' ∪ free_evars phi2))
∉ free_evars phi1'
evar_is_fresh_in
  (evar_fresh
     (elements (free_evars phi1' ∪ free_evars phi2)))
  phi1' auto.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ)
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}

        remember (update_evar_val (fresh_evar (phi1^[svar: dbi ↦ phi2])) c ρ) as ρe1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
eval ρe1'
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
        remember (update_evar_val Xu c ρ) as ρe2'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
eval ρe1'
  phi1^[svar:dbi↦phi2]^{evar:0↦fresh_evar
                                 phi1^[svar:dbi↦phi2]} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
        rewrite -> evar_open_bsvar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
eval ρe1'
  phi1^{evar:0↦fresh_evar phi1^[svar:dbi↦phi2]}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
well_formed_closed phi2 2: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
eval ρe1'
  phi1^{evar:0↦fresh_evar phi1^[svar:dbi↦phi2]}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
        remember (fresh_evar (phi1^[svar: dbi ↦ phi2])) as Xfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}

        (* dbi may or may not occur in phi1 *)
        remember (bsvar_occur phi1 dbi) as Hoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
Hoc: bool
HeqHoc: Hoc = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
        move: HeqHoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
Hoc: bool
Hoc = bsvar_occur phi1 dbi
→ eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
  eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
        case: Hoc => HeqHoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: true = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
        -- (* dbi ocurs in phi1 *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: true = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          pose proof (HXfr1Fresh := set_evar_fresh_is_fresh (phi1^[svar: dbi ↦ phi2])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: true = bsvar_occur phi1 dbi
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          rewrite <- HeqXfr1 in HXfr1Fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: true = bsvar_occur phi1 dbi
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          symmetry in HeqHoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          pose proof (Hsub := bsvar_subst_contains_subformula phi1 phi2 dbi HeqHoc).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          pose proof (HXfr1Fresh2 := evar_fresh_in_subformula _ _ _ Hsub HXfr1Fresh).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}

          assert (Hinterp:
                    (eval ρe1' phi2) =
                    (eval ρ phi2)
                 ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
eval ρe1' phi2 = eval ρ phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
eval ρe1' phi2 = eval ρ phi2 subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Hfresh_subst: evar_is_fresh_in
  (fresh_evar
     phi1^[svar:dbi↦patt_free_svar X])
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val
         (fresh_evar
            phi1^[svar:dbi↦
            patt_free_svar X]) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      fresh_evar
        phi1^[svar:dbi↦
        patt_free_svar X]} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
eval
  (update_evar_val (fresh_evar phi1^[svar:dbi↦phi2]) c
     ρ) phi2 = eval ρ phi2 apply eval_free_evar_independent.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Hfresh_subst: evar_is_fresh_in
  (fresh_evar
     phi1^[svar:dbi↦patt_free_svar X])
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val
         (fresh_evar
            phi1^[svar:dbi↦
            patt_free_svar X]) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      fresh_evar
        phi1^[svar:dbi↦
        patt_free_svar X]} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
evar_is_fresh_in (fresh_evar phi1^[svar:dbi↦phi2])
  phi2 auto. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}

          assert (He1e1':
                    (update_svar_val X (eval ρe1' phi2) ρ) =
                    (update_svar_val X (eval ρ phi2) ρ)
                 ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ congruence. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
          subst ρ2'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1'^{evar:0↦Xu}
          rewrite <- He1e1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρe1' phi2) ρ))
  phi1'^{evar:0↦Xu}
          subst phi1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρe1' phi2) ρ))
  phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
          rewrite -> evar_open_bsvar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρe1' phi2) ρ))
  phi1^{evar:0↦Xu}^[svar:dbi↦patt_free_svar X]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
well_formed_closed (patt_free_svar X)
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
well_formed_closed (patt_free_svar X) 
            wf_auto2.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρe1' phi2) ρ))
  phi1^{evar:0↦Xu}^[svar:dbi↦patt_free_svar X]

          assert (HXu: (Xfr1 = Xu)).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
Xfr1 = Xu
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρe1' phi2) ρ))
  phi1^{evar:0↦Xu}^[svar:dbi↦
  patt_free_svar X]
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
Xfr1 = Xu subst Xfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
fresh_evar phi1^[svar:dbi↦phi2] = Xu subst Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
fresh_evar phi1^[svar:dbi↦phi2] =
evar_fresh
  (elements
     (free_evars phi1^[svar:dbi↦patt_free_svar X]
      ∪ free_evars phi2)) unfold fresh_evar.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
evar_fresh_s (free_evars phi1^[svar:dbi↦phi2]) =
evar_fresh
  (elements
     (free_evars phi1^[svar:dbi↦patt_free_svar X]
      ∪ free_evars phi2))
            rewrite -> free_evars_bsvar_subst_eq.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
evar_fresh_s (free_evars phi1 ∪ free_evars phi2) =
evar_fresh
  (elements
     (free_evars phi1^[svar:dbi↦patt_free_svar X]
      ∪ free_evars phi2))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
bsvar_occur phi1 dbi
            fold (phi1^{svar: dbi ↦ X}).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
evar_fresh_s (free_evars phi1 ∪ free_evars phi2) =
evar_fresh
  (elements
     (free_evars phi1^{svar:dbi↦X} ∪ free_evars phi2))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
bsvar_occur phi1 dbi
            rewrite -> free_evars_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
evar_fresh_s (free_evars phi1 ∪ free_evars phi2) =
evar_fresh
  (elements (free_evars phi1 ∪ free_evars phi2))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
bsvar_occur phi1 dbi auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
ρe1': Valuation
Heqρe1': ρe1' =
update_evar_val
  (fresh_evar phi1^[svar:dbi↦phi2]) c ρ
ρe2': Valuation
Heqρe2': ρe2' =
update_evar_val
  (evar_fresh
     (elements
        (free_evars
           phi1^[svar:dbi↦
           patt_free_svar X]
         ∪ free_evars phi2))) c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2])
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in
  (fresh_evar phi1^[svar:dbi↦phi2]) phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
bsvar_occur phi1 dbi auto.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xfr1
  phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xfr1 phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρe1' phi2) ρ))
  phi1^{evar:0↦Xu}^[svar:dbi↦patt_free_svar X]
          rewrite -> HXu in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
eval ρe1' phi1^{evar:0↦Xu}^[svar:dbi↦phi2] =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρe1' phi2) ρ))
  phi1^{evar:0↦Xu}^[svar:dbi↦patt_free_svar X]
          (*
          assert (He1e2 : ρe1' = ρe2').
          { subst. auto. }
          rewrite <- He1e2. *)
          rewrite update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
eval ρe1' phi1^{evar:0↦Xu}^[svar:dbi↦phi2] =
eval
  (update_svar_val X (eval ρe1' phi2)
     (update_evar_val Xu c ρ))
  phi1^{evar:0↦Xu}^[svar:dbi↦patt_free_svar X]
          rewrite -Heqρe1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
eval ρe1' phi1^{evar:0↦Xu}^[svar:dbi↦phi2] =
eval (update_svar_val X (eval ρe1' phi2) ρe1')
  phi1^{evar:0↦Xu}^[svar:dbi↦patt_free_svar X]
          rewrite <- IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
eval ρe1' phi1^{evar:0↦Xu}^[svar:dbi↦phi2] =
eval ρe1' phi1^{evar:0↦Xu}^[svar:dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
size phi1^{evar:0↦Xu} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
well_formed_closed_mu_aux phi1^{evar:0↦Xu} (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
X ∉ free_svars phi1^{evar:0↦Xu}
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
eval ρe1' phi1^{evar:0↦Xu}^[svar:dbi↦phi2] =
eval ρe1' phi1^{evar:0↦Xu}^[svar:dbi↦phi2] reflexivity.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
size phi1^{evar:0↦Xu} ≤ sz rewrite <- evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
size phi1 ≤ sz lia. 
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
well_formed_closed phi2 auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
well_formed_closed_mu_aux phi1^{evar:0↦Xu} (S dbi) now apply wfc_mu_aux_body_ex_imp1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
X ∉ free_svars phi1^{evar:0↦Xu} rewrite -> free_svars_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xu = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xu c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = true
HXfr1Fresh: evar_is_fresh_in Xu phi1^[svar:dbi↦phi2]
Hsub: is_subformula_of_ind phi2 phi1^[svar:dbi↦phi2]
HXfr1Fresh2: evar_is_fresh_in Xu phi2
Hinterp: eval ρe1' phi2 = eval ρ phi2
He1e1': update_svar_val X (eval ρe1' phi2) ρ =
update_svar_val X (eval ρ phi2) ρ
HXu: Xfr1 = Xu
X ∉ free_svars phi1 auto.
          -- (* dbi does not occur in phi1 *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
            pose proof (HeqHoc' := HeqHoc).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc, HeqHoc': false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1}^[svar:dbi↦phi2] =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
            rewrite -> bsvar_subst_not_occur.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc, HeqHoc': false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc, HeqHoc': false = bsvar_occur phi1 dbi
well_formed_closed_mu_aux phi1^{evar:0↦Xfr1} dbi
            2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc, HeqHoc': false = bsvar_occur phi1 dbi
well_formed_closed_mu_aux phi1^{evar:0↦Xfr1} dbi apply wfc_mu_aux_body_ex_imp1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc, HeqHoc': false = bsvar_occur phi1 dbi
well_formed_closed_mu_aux phi1 dbi = true apply wfc_mu_lower in Hwfphi1; wf_auto2. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc, HeqHoc': false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
            (* Now svar_open does nothing to phi1, since it does not contain dbi (see HeqHoc). *)
            symmetry in HeqHoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (ex , phi1) (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu} simpl in Hwfphi1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 (S dbi)
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu} apply wfc_mu_lower in Hwfphi1; auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
eval ρe1' phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
            apply svar_open_not_occur with (x:=X) in Hwfphi1 as HWF.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
ρ2': Valuation
Heqρ2': ρ2' = update_svar_val X (eval ρ phi2) ρ
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval (update_evar_val Xfr2' c ρ2')
      phi1'^{evar:0↦Xfr2'} =
    eval (update_evar_val Xu c ρ2')
      phi1'^{evar:0↦Xu}
ρe1': Valuation
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
Heqρe1': ρe1' = update_evar_val Xfr1 c ρ
ρe2': Valuation
Heqρe2': ρe2' = update_evar_val Xu c ρ
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval ρe1' phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ2') phi1'^{evar:0↦Xu}
            (* X is not free in phi1, so the fact that in svar_val2' it is updated to some 
           value is irrelevant. *)
            subst ρ2' ρe1' ρe2'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval (update_evar_val Xfr1 c ρ) phi1^{evar:0↦Xfr1} =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1'^{evar:0↦Xu}
            replace (eval (update_evar_val Xu c (update_svar_val X (eval ρ phi2) ρ)) (phi1'^{evar: 0 ↦ Xu}))
            with (eval (update_evar_val Xu c ρ) (phi1'^{evar: 0 ↦ Xu})).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval (update_evar_val Xfr1 c ρ) phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ) phi1'^{evar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval (update_evar_val Xu c ρ) phi1'^{evar:0↦Xu} =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1'^{evar:0↦Xu}
            2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval (update_evar_val Xu c ρ) phi1'^{evar:0↦Xu} =
eval
  (update_evar_val Xu c
     (update_svar_val X (eval ρ phi2) ρ))
  phi1'^{evar:0↦Xu}
              rewrite update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval (update_evar_val Xu c ρ) phi1'^{evar:0↦Xu} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_evar_val Xu c ρ)) phi1'^{evar:0↦Xu}
              symmetry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval
  (update_svar_val X (eval ρ phi2)
     (update_evar_val Xu c ρ)) phi1'^{evar:0↦Xu} =
eval (update_evar_val Xu c ρ) phi1'^{evar:0↦Xu}
              apply eval_free_svar_independent.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
svar_is_fresh_in X phi1'^{evar:0↦Xu}
              unfold svar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
X ∉ free_svars phi1'^{evar:0↦Xu} rewrite -> free_svars_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
X ∉ free_svars phi1' subst phi1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
X ∉ free_svars phi1^[svar:dbi↦patt_free_svar X]
              fold (phi1^{svar: dbi ↦ X}).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
X ∉ free_svars phi1^{svar:dbi↦X}
              rewrite -> HWF.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
X ∉ free_svars phi1 auto.
            }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
phi1': Pattern
Heqphi1': phi1' = phi1^[svar:dbi↦patt_free_svar X]
Xfr2': evar
HeqXfr2': Xfr2' = fresh_evar phi1'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi1' ∪ free_evars phi2))
Hfresh_subst: evar_is_fresh_in Xfr2' phi1'
→ evar_is_fresh_in Xu phi1'
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1'^{evar:0↦Xu}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval (update_evar_val Xfr1 c ρ) phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ) phi1'^{evar:0↦Xu}
            subst phi1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^{svar:dbi↦X} = phi1
eval (update_evar_val Xfr1 c ρ) phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ)
  phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu} unfold svar_open in HWF.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
eval (update_evar_val Xfr1 c ρ) phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ)
  phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu} rewrite -> HWF.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
eval (update_evar_val Xfr1 c ρ) phi1^{evar:0↦Xfr1} =
eval (update_evar_val Xu c ρ) phi1^{evar:0↦Xu}
            rewrite -> eval_fresh_evar_open with (y := Xu).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
eval (update_evar_val Xu c ρ) phi1^{evar:0↦Xu} =
eval (update_evar_val Xu c ρ) phi1^{evar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in Xfr1 phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in Xu phi1
            {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
eval (update_evar_val Xu c ρ) phi1^{evar:0↦Xu} =
eval (update_evar_val Xu c ρ) phi1^{evar:0↦Xu} reflexivity. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in Xfr1 phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in Xu phi1
            {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in Xfr1 phi1 subst Xfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in (fresh_evar phi1^[svar:dbi↦phi2])
  phi1
              symmetry in HeqHoc'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
HeqHoc, HeqHoc': bsvar_occur phi1 dbi = false
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in (fresh_evar phi1^[svar:dbi↦phi2])
  phi1
              pose proof (Hsubst := bsvar_subst_not_occur dbi phi2 _ Hwfphi1).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
HeqHoc, HeqHoc': bsvar_occur phi1 dbi = false
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hsubst: phi1^[svar:dbi↦phi2] = phi1
evar_is_fresh_in (fresh_evar phi1^[svar:dbi↦phi2])
  phi1
              rewrite Hsubst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
HeqHoc, HeqHoc': bsvar_occur phi1 dbi = false
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hsubst: phi1^[svar:dbi↦phi2] = phi1
evar_is_fresh_in (fresh_evar phi1) phi1 auto.
            }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in Xu phi1
            {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
evar_is_fresh_in Xu phi1 pose proof (Hfr := set_evar_fresh_is_fresh').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hfr: ∀ S : EVarSet, evar_fresh (elements S) ∉ S
evar_is_fresh_in Xu phi1
              specialize (Hfr (union (free_evars (phi1^{svar: dbi ↦ X})) (free_evars phi2))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Xu: evar
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in Xu
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val Xu c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xu}
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars
        phi1^[svar:dbi↦
        patt_free_svar X] ∪ 
      free_evars phi2))
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hfr: evar_fresh
  (elements
     (free_evars phi1^{svar:dbi↦X}
      ∪ free_evars phi2))
∉ free_evars phi1^{svar:dbi↦X} ∪ free_evars phi2
evar_is_fresh_in Xu phi1
              subst Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hfr: evar_fresh
  (elements
     (free_evars phi1^{svar:dbi↦X}
      ∪ free_evars phi2))
∉ free_evars phi1^{svar:dbi↦X} ∪ free_evars phi2
evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi1^[svar:dbi↦patt_free_svar X]
         ∪ free_evars phi2))) phi1
              apply not_elem_of_union in Hfr.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hfr: (evar_fresh
   (elements
      (free_evars phi1^{svar:dbi↦X}
       ∪ free_evars phi2))
 ∉ free_evars phi1^{svar:dbi↦X})
∧ evar_fresh
    (elements
       (free_evars phi1^{svar:dbi↦X}
        ∪ free_evars phi2)) ∉ 
  free_evars phi2
evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi1^[svar:dbi↦patt_free_svar X]
         ∪ free_evars phi2))) phi1 destruct Hfr as [Hfr _].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hfr: evar_fresh
  (elements
     (free_evars phi1^{svar:dbi↦X}
      ∪ free_evars phi2))
∉ free_evars phi1^{svar:dbi↦X}
evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi1^[svar:dbi↦patt_free_svar X]
         ∪ free_evars phi2))) phi1
              rewrite free_evars_svar_open in Hfr.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hfr: evar_fresh
  (elements (free_evars phi1 ∪ free_evars phi2))
∉ free_evars phi1
evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi1^[svar:dbi↦patt_free_svar X]
         ∪ free_evars phi2))) phi1
              rewrite free_evars_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^[svar:dbi↦
              patt_free_svar X]
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux phi1 dbi = true
H: X ∉ free_svars phi1
c: M
Xfr2': evar
HeqXfr2': Xfr2' =
fresh_evar phi1^[svar:dbi↦patt_free_svar X]
Hfresh_subst: evar_is_fresh_in Xfr2'
  phi1^[svar:dbi↦patt_free_svar X]
→ evar_is_fresh_in
    (evar_fresh
       (elements
          (free_evars
             phi1^[svar:dbi↦
             patt_free_svar X]
           ∪ free_evars phi2)))
    phi1^[svar:dbi↦patt_free_svar X]
  → eval
      (update_evar_val Xfr2' c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦Xfr2'} =
    eval
      (update_evar_val
         (evar_fresh
            (elements
               (free_evars
                  phi1^[svar:dbi↦
                  patt_free_svar X]
                ∪ free_evars phi2))) c
         (update_svar_val X
            (eval ρ phi2) ρ))
      phi1^[svar:dbi↦patt_free_svar X]^{evar:0↦
      evar_fresh
        (elements
           (free_evars
              phi1^[svar:dbi↦
              patt_free_svar X]
            ∪ free_evars phi2))}
Xfr1: evar
HeqXfr1: Xfr1 = fresh_evar phi1^[svar:dbi↦phi2]
HeqHoc: bsvar_occur phi1 dbi = false
HeqHoc': false = bsvar_occur phi1 dbi
HWF: phi1^[svar:dbi↦patt_free_svar X] = phi1
Hfr: evar_fresh
  (elements (free_evars phi1 ∪ free_evars phi2))
∉ free_evars phi1
evar_is_fresh_in
  (evar_fresh
     (elements (free_evars phi1 ∪ free_evars phi2)))
  phi1 auto.
            }

      + (* Mu case *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
eval ρ (mu , phi1^[svar:S dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (mu , phi1)^{svar:dbi↦X}
        mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
eval ρ (mu , phi1^[svar:S dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (mu , phi1^{svar:increase_mu pm_spec_data dbi↦X}) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
eval ρ (mu , phi1^[svar:S dbi↦phi2]) =
eval (update_svar_val X (eval ρ phi2) ρ)
  (mu , phi1^{svar:S dbi↦X})
        rewrite 2!eval_mu_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
(let X := fresh_svar phi1^[svar:S dbi↦phi2] in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' := update_svar_val X S0 ρ in
      eval ρ' phi1^[svar:S dbi↦phi2]^{svar:0↦X})) =
(let X0 := fresh_svar phi1^{svar:S dbi↦X} in
 LeastFixpointOf
   (λ S0 : propset M,
      let ρ' :=
        update_svar_val X0 S0
          (update_svar_val X (eval ρ phi2) ρ) in
      eval ρ' phi1^{svar:S dbi↦X}^{svar:0↦X0})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val
          (fresh_svar phi1^[svar:S dbi↦phi2]) S0 ρ)
       phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                        phi1^[svar:
                                        S dbi↦phi2]}) =
LeastFixpointOf
  (λ S0 : propset M,
     eval
       (update_svar_val
          (fresh_svar phi1^{svar:S dbi↦X}) S0
          (update_svar_val X (eval ρ phi2) ρ))
       phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                     phi1^{svar:S dbi↦X}})
        apply f_equal.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
(λ S0 : propset M,
   eval
     (update_svar_val
        (fresh_svar phi1^[svar:S dbi↦phi2]) S0 ρ)
     phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                      phi1^[svar:S dbi↦phi2]}) =
(λ S0 : propset M,
   eval
     (update_svar_val (fresh_svar phi1^{svar:S dbi↦X})
        S0 (update_svar_val X (eval ρ phi2) ρ))
     phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                   phi1^{svar:S dbi↦X}}) apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
∀ x : propset M,
  eval
    (update_svar_val
       (fresh_svar phi1^[svar:S dbi↦phi2]) x ρ)
    phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                     phi1^[svar:S dbi↦phi2]} =
  eval
    (update_svar_val (fresh_svar phi1^{svar:S dbi↦X})
       x (update_svar_val X (eval ρ phi2) ρ))
    phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                  phi1^{svar:S dbi↦X}} intros E.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        remember (phi1^[svar: S dbi ↦ phi2]) as phi_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
        remember (union (union (union (union (union (free_svars phi_subst) (singleton (fresh_svar phi1))) (free_svars phi1)) (free_svars (phi1^{svar: S dbi ↦ X}))) (singleton X)) (free_svars phi2)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        remember (svar_fresh (elements B)) as Y.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
        assert (Hfreshy: Y <> fresh_svar phi1).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Y ≠ fresh_svar phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Y ≠ fresh_svar phi1 solve_fresh_svar_neq. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        assert (Hfreshy': svar_is_fresh_in Y phi1).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
svar_is_fresh_in Y phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
svar_is_fresh_in Y phi1
          subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]} ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi1
          eapply svar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi1 ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]} ∪ 
      free_svars phi1 ∪ 
      free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
      ∪ free_svars phi2)) ∉ 
?B
          2: apply set_svar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi1
⊆ free_svars phi1^[svar:S dbi↦phi2]
  ∪ {[fresh_svar phi1]} ∪ free_svars phi1
  ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
  ∪ free_svars phi2
          solve_free_svars_inclusion 5.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        assert (Hfreshy'': svar_is_fresh_in Y (phi1^{svar: S dbi ↦ X})).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
svar_is_fresh_in Y phi1^{svar:S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
svar_is_fresh_in Y phi1^{svar:S dbi↦X}
          subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]} ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi1^{svar:S dbi↦X}
          eapply svar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi1^{svar:S dbi↦X} ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]} ∪ 
      free_svars phi1 ∪ 
      free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
      ∪ free_svars phi2)) ∉ 
?B
          2: apply set_svar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi1^{svar:S dbi↦X}
⊆ free_svars phi1^[svar:S dbi↦phi2]
  ∪ {[fresh_svar phi1]} ∪ free_svars phi1
  ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
  ∪ free_svars phi2
          solve_free_svars_inclusion 5.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        assert (Hfreshy''': Y <> X).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Y ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Y ≠ X solve_fresh_svar_neq. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        assert (Hfreshy'''': svar_is_fresh_in Y phi2).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
svar_is_fresh_in Y phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
svar_is_fresh_in Y phi2
          subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]} ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
          eapply svar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi2 ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]} ∪ 
      free_svars phi1 ∪ 
      free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
      ∪ free_svars phi2)) ∉ 
?B
          2: apply set_svar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi2
⊆ free_svars phi1^[svar:S dbi↦phi2]
  ∪ {[fresh_svar phi1]} ∪ free_svars phi1
  ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
  ∪ free_svars phi2
          solve_free_svars_inclusion 5.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        assert (Hfreshy5: svar_is_fresh_in Y (phi1^[svar: S dbi ↦ phi2])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
          subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]} ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi1^[svar:S dbi↦phi2]
          eapply svar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi1^[svar:S dbi↦phi2] ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]} ∪ 
      free_svars phi1 ∪ 
      free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
      ∪ free_svars phi2)) ∉ 
?B
          2: apply set_svar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
free_svars phi1^[svar:S dbi↦phi2]
⊆ free_svars phi1^[svar:S dbi↦phi2]
  ∪ {[fresh_svar phi1]} ∪ free_svars phi1
  ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
  ∪ free_svars phi2
          solve_free_svars_inclusion 7.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

        remember (bsvar_occur phi1 (S dbi)) as Hoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
Hoc: bool
HeqHoc: Hoc = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
        destruct Hoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
        --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
          subst phi_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

          rewrite (eval_fresh_svar_open _ ((fresh_svar (phi1^{svar: S dbi ↦ X}))) Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in (fresh_svar phi1^{svar:S dbi↦X})
  phi1^{svar:S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1^{svar:S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦
  fresh_svar phi1^[svar:
    S dbi↦phi2]} =
eval
  (update_svar_val Y E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦Y}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in (fresh_svar phi1^{svar:S dbi↦X})
  phi1^{svar:S dbi↦X} apply set_svar_fresh_is_fresh. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1^{svar:S dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦
  fresh_svar phi1^[svar:
    S dbi↦phi2]} =
eval
  (update_svar_val Y E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦Y}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1^{svar:S dbi↦X} apply Hfreshy''. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val Y E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦Y}

          rewrite update_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
Y ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦
  fresh_svar phi1^[svar:
    S dbi↦phi2]} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_svar_val Y E ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦Y}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
Y ≠ X apply Hfreshy'''. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_svar_val Y E ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦Y}
          rewrite svar_open_comm_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦
  fresh_svar phi1^[svar:
    S dbi↦phi2]} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_svar_val Y E ρ))
  phi1^{svar:0↦Y}^{svar:
  Init.Nat.pred (S dbi)↦X}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_svar_val Y E ρ))
  phi1^{svar:0↦Y}^{svar:Init.Nat.pred (S dbi)↦X}
          replace (pred (S dbi)) with dbi by lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_svar_val Y E ρ))
  phi1^{svar:0↦Y}^{svar:dbi↦X}

          replace (eval ρ phi2) with (eval (update_svar_val Y E ρ) phi2).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val X
     (eval (update_svar_val Y E ρ) phi2)
     (update_svar_val Y E ρ))
  phi1^{svar:0↦Y}^{svar:dbi↦X}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi2 = eval ρ phi2
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi2 = eval ρ phi2
            apply eval_free_svar_independent.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi2
            apply Hfreshy''''.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val X
     (eval (update_svar_val Y E ρ) phi2)
     (update_svar_val Y E ρ))
  phi1^{svar:0↦Y}^{svar:dbi↦X}

          rewrite <- IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
size phi1^{svar:0↦Y} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1^{svar:0↦Y} (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X ∉ free_svars phi1^{svar:0↦Y}
          4: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1^{svar:0↦Y} (S dbi) apply wfc_mu_aux_body_mu_imp3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
0 ≤ S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S (S dbi)) = true lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S (S dbi)) = true auto. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
size phi1^{svar:0↦Y} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X ∉ free_svars phi1^{svar:0↦Y}
          4: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X ∉ free_svars phi1^{svar:0↦Y} apply svar_is_fresh_in_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X ≠ Y
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in X phi1
               apply not_eq_sym.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
Y ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in X phi1
               apply Hfreshy'''.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in X phi1
               simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars phi1
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in X phi1
               apply H.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
size phi1^{svar:0↦Y} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed phi2
          3: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed phi2 apply Hwfc2. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
size phi1^{svar:0↦Y} ≤ sz
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
size phi1^{svar:0↦Y} ≤ sz rewrite -svar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
size phi1 ≤ sz simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
size phi1 ≤ sz lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]

          rewrite svar_open_bsvar_subst_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^{svar:0↦fresh_svar phi1^[svar:S dbi↦phi2]}^[svar:
  Init.Nat.pred (S dbi)↦phi2] =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
well_formed_closed phi2 apply Hwfc2. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^{svar:0↦fresh_svar phi1^[svar:S dbi↦phi2]}^[svar:
  Init.Nat.pred (S dbi)↦phi2] =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^{svar:0↦fresh_svar phi1^[svar:S dbi↦phi2]}^[svar:
  Init.Nat.pred (S dbi)↦phi2] =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
          remember  (fresh_svar (phi1^[svar: S dbi ↦ phi2])) as X'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^{svar:0↦X'}^[svar:Init.Nat.pred (S dbi)↦phi2] =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]

          rewrite -svar_open_bsvar_subst_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
well_formed_closed phi2 apply Hwfc2. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:dbi↦phi2]
          replace dbi with (pred (S dbi)) by lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S (Init.Nat.pred (S dbi))↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^{svar:0↦Y}^[svar:Init.Nat.pred (S dbi)↦phi2]
          rewrite <- svar_open_bsvar_subst_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S (Init.Nat.pred (S dbi))↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
0 < S dbi
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
well_formed_closed phi2 apply Hwfc2. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S (Init.Nat.pred (S dbi))↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦Y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
0 < S dbi
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S (Init.Nat.pred (S dbi))↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦Y}
          replace (pred (S dbi)) with dbi by lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
eval (update_svar_val X' E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦X'} =
eval (update_svar_val Y E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦Y}
          apply eval_fresh_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
svar_is_fresh_in X' phi1^[svar:S dbi↦phi2]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
svar_is_fresh_in X' phi1^[svar:S dbi↦phi2] subst X'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
svar_is_fresh_in (fresh_svar phi1^[svar:S dbi↦phi2])
  phi1^[svar:S dbi↦phi2] apply set_svar_fresh_is_fresh. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: true = bsvar_occur phi1 (S dbi)
X': svar
HeqX': X' = fresh_svar phi1^[svar:S dbi↦phi2]
svar_is_fresh_in Y phi1^[svar:S dbi↦phi2] apply Hfreshy5.  }

        --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi_subst) E ρ)
  phi_subst^{svar:0↦fresh_svar phi_subst} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
          rewrite Heqphi_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval
  (update_svar_val (fresh_svar phi1^[svar:S dbi↦phi2])
     E ρ)
  phi1^[svar:S dbi↦phi2]^{svar:0↦fresh_svar
                                   phi1^[svar:S dbi↦phi2]} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
          rewrite bsvar_subst_not_occur.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}} auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
          rewrite bsvar_subst_not_occur in Heqphi_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
Heqphi_subst: phi_subst = phi1
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}} all: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
Heqphi_subst: phi_subst = phi1
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦
  fresh_svar phi1^{svar:
    S dbi↦X}}
          1-2: apply wfc_mu_lower; auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

          rewrite -> eval_fresh_svar_open with (X := fresh_svar phi1) (Y := Y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in (fresh_svar phi1) phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in (fresh_svar phi1) phi1 apply set_svar_fresh_is_fresh. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1 subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy5: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^[svar:S dbi↦phi2]
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]} ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi1
               eapply svar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy5: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^[svar:S dbi↦phi2]
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
HeqHoc: false = bsvar_occur phi1 (S dbi)
free_svars phi1 ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy5: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^[svar:S dbi↦phi2]
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]} ∪ 
      free_svars phi1 ∪ 
      free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
      ∪ free_svars phi2)) ∉ 
?B
               2: apply set_svar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
Hfreshy5: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^[svar:S dbi↦phi2]
Hfreshy'''': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars
           phi1^[svar:
           S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars
             phi1^{svar:
             S dbi↦X} ∪ {[X]}
         ∪ free_svars phi2))) phi2
Hfreshy''': svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2)) ≠ X
Hfreshy'': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2)))
  phi1^{svar:S dbi↦X}
Hfreshy': svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi1^[svar:S dbi↦phi2]
         ∪ {[fresh_svar phi1]}
         ∪ free_svars phi1
         ∪ free_svars phi1^{svar:S dbi↦X}
         ∪ {[X]} ∪ free_svars phi2))) phi1
Hfreshy: svar_fresh
  (elements
     (free_svars phi1^[svar:S dbi↦phi2]
      ∪ {[fresh_svar phi1]}
      ∪ free_svars phi1
      ∪ free_svars phi1^{svar:S dbi↦X}
      ∪ {[X]} ∪ free_svars phi2))
≠ fresh_svar phi1
HeqHoc: false = bsvar_occur phi1 (S dbi)
free_svars phi1
⊆ free_svars phi1^[svar:S dbi↦phi2]
  ∪ {[fresh_svar phi1]} ∪ free_svars phi1
  ∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
  ∪ free_svars phi2
               solve_free_svars_inclusion 5.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: size (mu , phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

          simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
phi_subst: Pattern
Heqphi_subst: phi_subst = phi1^[svar:S dbi↦phi2]
B: SVarSet
HeqB: B =
free_svars phi_subst ∪ {[fresh_svar phi1]}
∪ free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

          subst phi_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1^{svar:S dbi↦X}) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:S dbi↦X}^{svar:0↦fresh_svar
                                phi1^{svar:S dbi↦X}}

          rewrite -> svar_open_not_occur with (x := X)(n := S dbi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:0↦fresh_svar phi1}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S dbi)
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
well_formed_closed_mu_aux phi1 (S dbi) apply wfc_mu_lower; auto. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:0↦fresh_svar phi1}

          destruct (decide (X = (fresh_svar phi1))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
e: X = fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:0↦fresh_svar phi1}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:0↦fresh_svar phi1}
          ++
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
e: X = fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:0↦fresh_svar phi1} subst X.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: fresh_svar phi1 ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦fresh_svar phi1}
∪ {[fresh_svar phi1]} ∪ 
free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy''': Y ≠ fresh_svar phi1
Hfreshy'': svar_is_fresh_in Y
  phi1^{svar:S dbi↦fresh_svar phi1}
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1) E
     (update_svar_val (fresh_svar phi1) (eval ρ phi2)
        ρ)) phi1^{svar:0↦fresh_svar phi1} rewrite update_svar_val_shadow.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: fresh_svar phi1 ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦fresh_svar phi1}
∪ {[fresh_svar phi1]} ∪ 
free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy''': Y ≠ fresh_svar phi1
Hfreshy'': svar_is_fresh_in Y
  phi1^{svar:S dbi↦fresh_svar phi1}
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1}
             apply eval_fresh_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: fresh_svar phi1 ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦fresh_svar phi1}
∪ {[fresh_svar phi1]} ∪ 
free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy''': Y ≠ fresh_svar phi1
Hfreshy'': svar_is_fresh_in Y
  phi1^{svar:S dbi↦fresh_svar phi1}
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: fresh_svar phi1 ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦fresh_svar phi1}
∪ {[fresh_svar phi1]} ∪ 
free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy''': Y ≠ fresh_svar phi1
Hfreshy'': svar_is_fresh_in Y
  phi1^{svar:S dbi↦fresh_svar phi1}
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in (fresh_svar phi1) phi1
             {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: fresh_svar phi1 ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦fresh_svar phi1}
∪ {[fresh_svar phi1]} ∪ 
free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy''': Y ≠ fresh_svar phi1
Hfreshy'': svar_is_fresh_in Y
  phi1^{svar:S dbi↦fresh_svar phi1}
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in Y phi1 apply Hfreshy'. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: fresh_svar phi1 ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦fresh_svar phi1}
∪ {[fresh_svar phi1]} ∪ 
free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy''': Y ≠ fresh_svar phi1
Hfreshy'': svar_is_fresh_in Y
  phi1^{svar:S dbi↦fresh_svar phi1}
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
svar_is_fresh_in (fresh_svar phi1) phi1
             apply set_svar_fresh_is_fresh.
          ++
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val (fresh_svar phi1) E
     (update_svar_val X (eval ρ phi2) ρ))
  phi1^{svar:0↦fresh_svar phi1} rewrite update_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
fresh_svar phi1 ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_svar_val (fresh_svar phi1) E ρ))
  phi1^{svar:0↦fresh_svar phi1}
             {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
fresh_svar phi1 ≠ X apply not_eq_sym.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
X ≠ fresh_svar phi1 apply n. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval
  (update_svar_val X (eval ρ phi2)
     (update_svar_val (fresh_svar phi1) E ρ))
  phi1^{svar:0↦fresh_svar phi1}
             rewrite -> eval_free_svar_independent with (X:=X).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in X phi1^{svar:0↦fresh_svar phi1}
             2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in X phi1^{svar:0↦fresh_svar phi1} apply svar_is_fresh_in_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
X ≠ fresh_svar phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in X phi1 apply n.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in X phi1
                  simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars phi1
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in X phi1 apply H.
             }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
eval (update_svar_val Y E ρ) phi1^{svar:0↦Y} =
eval (update_svar_val (fresh_svar phi1) E ρ)
  phi1^{svar:0↦fresh_svar phi1}
             apply eval_fresh_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in Y phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in (fresh_svar phi1) phi1
             {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in Y phi1 apply Hfreshy'. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi1 phi2 : Pattern),
  size phi1 ≤ sz
  → ∀ (ρ : Valuation) (X : svar),
      well_formed_closed phi2
      → well_formed_closed_mu_aux phi1 (S dbi)
        → X ∉ free_svars phi1
          → eval ρ phi1^[svar:dbi↦phi2] =
            eval
              (update_svar_val X 
                 (eval ρ phi2) ρ)
              phi1^{svar:dbi↦X}
dbi: db_index
phi1, phi2: Pattern
Hsz: S (size phi1) ≤ S sz
ρ: Valuation
X: svar
Hwfc2: well_formed_closed phi2
Hwfphi1: well_formed_closed_mu_aux 
  (mu , phi1) (S dbi)
H: X ∉ free_svars (mu , phi1)
E: propset M
B: SVarSet
HeqB: B =
free_svars phi1^[svar:S dbi↦phi2]
∪ {[fresh_svar phi1]} ∪ 
free_svars phi1
∪ free_svars phi1^{svar:S dbi↦X} ∪ {[X]}
∪ free_svars phi2
Y: svar
HeqY: Y = svar_fresh (elements B)
Hfreshy: Y ≠ fresh_svar phi1
Hfreshy': svar_is_fresh_in Y phi1
Hfreshy'': svar_is_fresh_in Y phi1^{svar:S dbi↦X}
Hfreshy''': Y ≠ X
Hfreshy'''': svar_is_fresh_in Y phi2
Hfreshy5: svar_is_fresh_in Y phi1^[svar:S dbi↦phi2]
HeqHoc: false = bsvar_occur phi1 (S dbi)
n: X ≠ fresh_svar phi1
svar_is_fresh_in (fresh_svar phi1) phi1
             apply set_svar_fresh_is_fresh.
  Qed.

  Lemma set_substitution_lemma : forall (dbi : db_index) (phi1 phi2 : Pattern),
      forall (ρ : Valuation) (X : svar),
        well_formed_closed phi2 -> well_formed_closed_mu_aux phi1 (S dbi) ->
        ~ elem_of X (free_svars phi1) ->
        @eval M ρ (phi1^[svar: dbi ↦ phi2])
        = eval (update_svar_val X (@eval M ρ phi2) ρ) (phi1^{svar: dbi ↦ X}).
signature: Signature
M: Model
∀ (dbi : db_index) (phi1 phi2 : Pattern) (ρ : Valuation) 
  (X : svar),
  well_formed_closed phi2
  → well_formed_closed_mu_aux phi1 (S dbi)
    → X ∉ free_svars phi1
      → eval ρ phi1^[svar:dbi↦phi2] =
        eval (update_svar_val X (eval ρ phi2) ρ)
          phi1^{svar:dbi↦X}
  Proof.
signature: Signature
M: Model
∀ (dbi : db_index) (phi1 phi2 : Pattern) (ρ : Valuation) 
  (X : svar),
  well_formed_closed phi2
  → well_formed_closed_mu_aux phi1 (S dbi)
    → X ∉ free_svars phi1
      → eval ρ phi1^[svar:dbi↦phi2] =
        eval (update_svar_val X (eval ρ phi2) ρ)
          phi1^{svar:dbi↦X}
    intros.
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
eval ρ phi1^[svar:dbi↦phi2] =
eval (update_svar_val X (eval ρ phi2) ρ)
  phi1^{svar:dbi↦X}
    apply Private_plugging_patterns with (sz := size phi1).
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
size phi1 ≤ size phi1
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
well_formed_closed phi2
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
X ∉ free_svars phi1
    lia.
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
well_formed_closed phi2
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
well_formed_closed_mu_aux phi1 (S dbi)
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
ρ: Valuation
X: svar
H: well_formed_closed phi2
H0: well_formed_closed_mu_aux phi1 (S dbi)
H1: X ∉ free_svars phi1
X ∉ free_svars phi1 all: auto.
  Qed.

  Lemma Private_plugging_patterns_bevar_subst : forall (sz : nat) (dbi : db_index) 
    (phi : Pattern) (y : evar),
      size phi <= sz -> forall (ρ : Valuation) (x : evar),
        evar_is_fresh_in x phi -> well_formed_closed_ex_aux phi (S dbi) ->
        @eval M ρ (phi^[evar: dbi ↦ patt_free_evar y])
        = eval (update_evar_val x (evar_valuation ρ y) ρ) (phi^{evar: dbi ↦ x}).
signature: Signature
M: Model
∀ (sz : nat) (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ phi^[evar:dbi↦patt_free_evar y] =
          eval
            (update_evar_val x (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
  Proof.
signature: Signature
M: Model
∀ (sz : nat) (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ phi^[evar:dbi↦patt_free_evar y] =
          eval
            (update_evar_val x (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
    induction sz; intros dbi phi y Hsz ρ x H Hwf.
signature: Signature
M: Model
dbi: db_index
phi: Pattern
y: evar
Hsz: size phi ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x phi
Hwf: well_formed_closed_ex_aux phi (S dbi)
eval ρ phi^[evar:dbi↦patt_free_evar y] =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  phi^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size phi ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x phi
Hwf: well_formed_closed_ex_aux phi (S dbi)
eval ρ phi^[evar:dbi↦patt_free_evar y] =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  phi^{evar:dbi↦x}
    - (* sz == 0 *)
signature: Signature
M: Model
dbi: db_index
phi: Pattern
y: evar
Hsz: size phi ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x phi
Hwf: well_formed_closed_ex_aux phi (S dbi)
eval ρ phi^[evar:dbi↦patt_free_evar y] =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  phi^{evar:dbi↦x}
      destruct phi; simpl in Hsz; simpl.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)^{evar:dbi↦x}
signature: Signature
M: Model
dbi: db_index
x0: svar
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_svar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_svar x0) 
  (S dbi)
eval ρ (patt_free_svar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_svar x0)^{evar:dbi↦x}
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar y
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n)^{evar:dbi↦x}
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_svar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_svar n) 
  (S dbi)
eval ρ (patt_bound_svar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_svar n)^{evar:dbi↦x}
signature: Signature
M: Model
dbi: db_index
sigma: symbols
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_sym sigma)
Hwf: well_formed_closed_ex_aux 
  (patt_sym sigma) (S dbi)
eval ρ (patt_sym sigma) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_sym sigma)^{evar:dbi↦x}
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 $ phi2)
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 $ phi2)^{evar:dbi↦x}
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x ⊥
Hwf: well_formed_closed_ex_aux ⊥ (S dbi)
eval ρ ⊥ =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  ⊥^{evar:dbi↦x}
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 ---> phi2)
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 ---> phi2)^{evar:dbi↦x}
signature: Signature
M: Model
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi)^{evar:dbi↦x}
signature: Signature
M: Model
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
eval ρ (mu , phi^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (mu , phi)^{evar:dbi↦x}
      + (* free_evar *)
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)^{evar:dbi↦x}
        unfold evar_is_fresh_in in H.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)^{evar:dbi↦x}
        cbn.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)
        repeat rewrite -> eval_free_evar_simpl.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[evar_valuation ρ x0]} =
{[evar_valuation
    (update_evar_val x (evar_valuation ρ y) ρ) x0]}
        unfold update_evar_val.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[evar_valuation ρ x0]} =
{[evar_valuation
    {|
      evar_valuation :=
        λ ev' : evar,
          if decide (x = ev')
          then evar_valuation ρ y
          else evar_valuation ρ ev';
      svar_valuation := svar_valuation ρ
    |} x0]}
        destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[evar_valuation
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
    x0]} =
{[evar_valuation
    {|
      evar_valuation :=
        λ ev' : evar,
          if decide (x = ev')
          then
           evar_valuation
             {|
               evar_valuation := ρₑ;
               svar_valuation := ρₛ
             |} y
          else
           evar_valuation
             {|
               evar_valuation := ρₑ;
               svar_valuation := ρₛ
             |} ev';
      svar_valuation :=
        svar_valuation
          {|
            evar_valuation := ρₑ; svar_valuation := ρₛ
          |}
    |} x0]} simpl.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[ρₑ x0]} =
{[if decide (x = x0) then ρₑ y else ρₑ x0]}
        destruct (decide (x = x0)).
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]}
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
n: x ≠ x0
{[ρₑ x0]} = {[ρₑ x0]}
        *
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]} simpl.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]} unfold not in H.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]} exfalso.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
False apply H.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
x ∈ free_evars (patt_free_evar x0)
          apply elem_of_singleton_2.
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
x = x0 auto.
        *
signature: Signature
M: Model
dbi: db_index
x0, y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
n: x ≠ x0
{[ρₑ x0]} = {[ρₑ x0]} auto.
      + (* free_svar *)
signature: Signature
M: Model
dbi: db_index
x0: svar
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_svar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_svar x0) 
  (S dbi)
eval ρ (patt_free_svar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_svar x0)^{evar:dbi↦x}
        cbn.
signature: Signature
M: Model
dbi: db_index
x0: svar
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_svar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_svar x0) 
  (S dbi)
eval ρ (patt_free_svar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_svar x0)
        auto.
      + (* bound_evar *)
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar y
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n)^{evar:dbi↦x}
        cbn.
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar y
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
        case_match; subst.
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_evar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n)
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar dbi) 
  (S dbi)
H: evar_is_fresh_in x (patt_bound_evar dbi)
H0: compare_nat dbi dbi = Nat_equal dbi dbi erefl
eval ρ (patt_free_evar y) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x)
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_evar (Nat.pred n)) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar (Nat.pred n))
        *
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_evar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n) auto.
        *
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar dbi) 
  (S dbi)
H: evar_is_fresh_in x (patt_bound_evar dbi)
H0: compare_nat dbi dbi = Nat_equal dbi dbi erefl
eval ρ (patt_free_evar y) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x) do 2 rewrite eval_free_evar_simpl.
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar dbi) 
  (S dbi)
H: evar_is_fresh_in x (patt_bound_evar dbi)
H0: compare_nat dbi dbi = Nat_equal dbi dbi erefl
{[evar_valuation ρ y]} =
{[evar_valuation
    (update_evar_val x (evar_valuation ρ y) ρ) x]}
          destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar dbi) 
  (S dbi)
H: evar_is_fresh_in x (patt_bound_evar dbi)
H0: compare_nat dbi dbi = Nat_equal dbi dbi erefl
{[evar_valuation
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |} y]} =
{[evar_valuation
    (update_evar_val x
       (evar_valuation
          {|
            evar_valuation := ρₑ; svar_valuation := ρₛ
          |} y)
       {|
         evar_valuation := ρₑ; svar_valuation := ρₛ
       |}) x]} simpl.
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar dbi) 
  (S dbi)
H: evar_is_fresh_in x (patt_bound_evar dbi)
H0: compare_nat dbi dbi = Nat_equal dbi dbi erefl
{[ρₑ y]} = {[if decide (x = x) then ρₑ y else ρₑ x]}
          unfold update_evar_val.
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar dbi) 
  (S dbi)
H: evar_is_fresh_in x (patt_bound_evar dbi)
H0: compare_nat dbi dbi = Nat_equal dbi dbi erefl
{[ρₑ y]} = {[if decide (x = x) then ρₑ y else ρₑ x]}
          case_match; auto.
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar dbi) 
  (S dbi)
H: evar_is_fresh_in x (patt_bound_evar dbi)
H0: compare_nat dbi dbi = Nat_equal dbi dbi erefl
n: x ≠ x
H1: decide (x = x) = right n
{[ρₑ y]} = {[ρₑ x]} contradiction.
        *
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_evar (Nat.pred n)) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar (Nat.pred n)) auto.
      + (* bound_svar *)
signature: Signature
M: Model
dbi, n: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_svar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_svar n) 
  (S dbi)
eval ρ (patt_bound_svar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_svar n)^{evar:dbi↦x}
        rewrite 2!eval_bound_svar_simpl; auto.
      + (* sym *)
signature: Signature
M: Model
dbi: db_index
sigma: symbols
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_sym sigma)
Hwf: well_formed_closed_ex_aux 
  (patt_sym sigma) (S dbi)
eval ρ (patt_sym sigma) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_sym sigma)^{evar:dbi↦x}
        simpl.
signature: Signature
M: Model
dbi: db_index
sigma: symbols
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_sym sigma)
Hwf: well_formed_closed_ex_aux 
  (patt_sym sigma) (S dbi)
eval ρ (patt_sym sigma) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_sym sigma)^{evar:dbi↦x} repeat rewrite -> eval_sym_simpl; auto.
      + (* app *)
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 $ phi2)
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 $ phi2)^{evar:dbi↦x}
        lia.
      + (* bot *)
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x ⊥
Hwf: well_formed_closed_ex_aux ⊥ (S dbi)
eval ρ ⊥ =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  ⊥^{evar:dbi↦x}
        simpl.
signature: Signature
M: Model
dbi: db_index
y: evar
Hsz: 0 ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x ⊥
Hwf: well_formed_closed_ex_aux ⊥ (S dbi)
eval ρ ⊥ =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  ⊥^{evar:dbi↦x} repeat rewrite eval_bott_simpl; auto.
      + (* impl *)
signature: Signature
M: Model
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 ---> phi2)
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 ---> phi2)^{evar:dbi↦x}
        lia.
      + (* ex *)
signature: Signature
M: Model
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi)^{evar:dbi↦x}
        lia.
      + (* mu *)
signature: Signature
M: Model
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ 0
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
eval ρ (mu , phi^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (mu , phi)^{evar:dbi↦x}
        lia.
    - (* sz = S sz' *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size phi ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x phi
Hwf: well_formed_closed_ex_aux phi (S dbi)
eval ρ phi^[evar:dbi↦patt_free_evar y] =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  phi^{evar:dbi↦x}
      destruct phi; simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0: svar
y: evar
Hsz: size (patt_free_svar x0) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_svar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_svar x0) 
  (S dbi)
eval ρ (patt_free_svar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_svar x0)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar y
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_svar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_svar n) 
  (S dbi)
eval ρ (patt_bound_svar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_svar n)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
sigma: symbols
y: evar
Hsz: size (patt_sym sigma) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_sym sigma)
Hwf: well_formed_closed_ex_aux 
  (patt_sym sigma) (S dbi)
eval ρ (patt_sym sigma) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_sym sigma)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 $ phi2)
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 $ phi2)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
y: evar
Hsz: size ⊥ ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x ⊥
Hwf: well_formed_closed_ex_aux ⊥ (S dbi)
eval ρ ⊥ =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  ⊥^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 ---> phi2) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 ---> phi2)
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 ---> phi2)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (ex , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi)^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
eval ρ (mu , phi^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (mu , phi)^{evar:dbi↦x}
      (* HERE we duplicate some of the effort. I do not like it. *)
      + (* free_evar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)^{evar:dbi↦x}
        unfold evar_is_fresh_in in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)^{evar:dbi↦x} cbn.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
eval ρ (patt_free_evar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x0)
        do 2 rewrite -> eval_free_evar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[evar_valuation ρ x0]} =
{[evar_valuation
    (update_evar_val x (evar_valuation ρ y) ρ) x0]}
        unfold update_evar_val.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[evar_valuation ρ x0]} =
{[evar_valuation
    {|
      evar_valuation :=
        λ ev' : evar,
          if decide (x = ev')
          then evar_valuation ρ y
          else evar_valuation ρ ev';
      svar_valuation := svar_valuation ρ
    |} x0]}
        destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[evar_valuation
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
    x0]} =
{[evar_valuation
    {|
      evar_valuation :=
        λ ev' : evar,
          if decide (x = ev')
          then
           evar_valuation
             {|
               evar_valuation := ρₑ;
               svar_valuation := ρₛ
             |} y
          else
           evar_valuation
             {|
               evar_valuation := ρₑ;
               svar_valuation := ρₛ
             |} ev';
      svar_valuation :=
        svar_valuation
          {|
            evar_valuation := ρₑ; svar_valuation := ρₛ
          |}
    |} x0]} simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
{[ρₑ x0]} =
{[if decide (x = x0) then ρₑ y else ρₑ x0]}
        destruct (decide (x = x0)).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
n: x ≠ x0
{[ρₑ x0]} = {[ρₑ x0]}
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]} simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]} unfold not in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
{[ρₑ x0]} = {[ρₑ y]} exfalso.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
False apply H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
x ∈ free_evars (patt_free_evar x0)
          apply elem_of_singleton_2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∈ free_evars (patt_free_evar x0) → False
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
e: x = x0
x = x0 auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0, y: evar
Hsz: size (patt_free_evar x0) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: x ∉ free_evars (patt_free_evar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_evar x0) 
  (S dbi)
n: x ≠ x0
{[ρₑ x0]} = {[ρₑ x0]} auto.
      + (* free_svar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0: svar
y: evar
Hsz: size (patt_free_svar x0) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_svar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_svar x0) 
  (S dbi)
eval ρ (patt_free_svar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_svar x0)^{evar:dbi↦x}
        cbn.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
x0: svar
y: evar
Hsz: size (patt_free_svar x0) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_free_svar x0)
Hwf: well_formed_closed_ex_aux 
  (patt_free_svar x0) 
  (S dbi)
eval ρ (patt_free_svar x0) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_svar x0)
        auto.
      + (* bound_evar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar y
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n)^{evar:dbi↦x}
        cbn.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
eval ρ
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar y
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
        case_match.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_evar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ (patt_free_evar y) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_evar (Nat.pred n)) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar (Nat.pred n))
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
l: n < dbi
H0: compare_nat n dbi = Nat_less n dbi l
eval ρ (patt_bound_evar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar n) auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
eval ρ (patt_free_evar y) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_free_evar x) repeat rewrite eval_free_evar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
{[evar_valuation ρ y]} =
{[evar_valuation
    (update_evar_val x (evar_valuation ρ y) ρ) x]} unfold update_evar_val.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
{[evar_valuation ρ y]} =
{[evar_valuation
    {|
      evar_valuation :=
        λ ev' : evar,
          if decide (x = ev')
          then evar_valuation ρ y
          else evar_valuation ρ ev';
      svar_valuation := svar_valuation ρ
    |} x]}
          destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
{[evar_valuation
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |} y]} =
{[evar_valuation
    {|
      evar_valuation :=
        λ ev' : evar,
          if decide (x = ev')
          then
           evar_valuation
             {|
               evar_valuation := ρₑ;
               svar_valuation := ρₛ
             |} y
          else
           evar_valuation
             {|
               evar_valuation := ρₑ;
               svar_valuation := ρₛ
             |} ev';
      svar_valuation :=
        svar_valuation
          {|
            evar_valuation := ρₑ; svar_valuation := ρₛ
          |}
    |} x]} simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
{[ρₑ y]} = {[if decide (x = x) then ρₑ y else ρₑ x]}
          destruct (decide (x = x)).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
e0: x = x
{[ρₑ y]} = {[ρₑ y]}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
n0: x ≠ x
{[ρₑ y]} = {[ρₑ x]} auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
n0: x ≠ x
{[ρₑ y]} = {[ρₑ x]} contradiction.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_evar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_evar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_evar n) 
  (S dbi)
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
eval ρ (patt_bound_evar (Nat.pred n)) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_evar (Nat.pred n)) auto.
      + (* bound_svar *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi, n: db_index
y: evar
Hsz: size (patt_bound_svar n) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_bound_svar n)
Hwf: well_formed_closed_ex_aux 
  (patt_bound_svar n) 
  (S dbi)
eval ρ (patt_bound_svar n) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_bound_svar n)^{evar:dbi↦x}
        rewrite 2!eval_bound_svar_simpl; auto.
      + (* sym *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
sigma: symbols
y: evar
Hsz: size (patt_sym sigma) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_sym sigma)
Hwf: well_formed_closed_ex_aux 
  (patt_sym sigma) (S dbi)
eval ρ (patt_sym sigma) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_sym sigma)^{evar:dbi↦x}
        simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
sigma: symbols
y: evar
Hsz: size (patt_sym sigma) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (patt_sym sigma)
Hwf: well_formed_closed_ex_aux 
  (patt_sym sigma) (S dbi)
eval ρ (patt_sym sigma) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (patt_sym sigma)^{evar:dbi↦x} repeat rewrite -> eval_sym_simpl; auto.
      (* HERE the duplication ends *)
      + (* app *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 $ phi2)
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 $ phi2)^{evar:dbi↦x}
        mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 $ phi2)
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} $ phi2^{evar:dbi↦x})
        unfold evar_is_fresh_in in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars (phi1 $ phi2)
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} $ phi2^{evar:dbi↦x}) simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1 ∪ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} $ phi2^{evar:dbi↦x}) apply not_elem_of_union in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: (x ∉ free_evars phi1) ∧ x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} $ phi2^{evar:dbi↦x}) destruct H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] $
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} $ phi2^{evar:dbi↦x})
        repeat rewrite -> eval_app_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 $ phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
app_ext (eval ρ phi1^[evar:dbi↦patt_free_evar y])
  (eval ρ phi2^[evar:dbi↦patt_free_evar y]) =
app_ext
  (eval (update_evar_val x (evar_valuation ρ y) ρ)
     phi1^{evar:dbi↦x})
  (eval (update_evar_val x (evar_valuation ρ y) ρ)
     phi2^{evar:dbi↦x})
        simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
app_ext (eval ρ phi1^[evar:dbi↦patt_free_evar y])
  (eval ρ phi2^[evar:dbi↦patt_free_evar y]) =
app_ext
  (eval (update_evar_val x (evar_valuation ρ y) ρ)
     phi1^{evar:dbi↦x})
  (eval (update_evar_val x (evar_valuation ρ y) ρ)
     phi2^{evar:dbi↦x})
        repeat rewrite <- IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
app_ext (eval ρ phi1^[evar:dbi↦patt_free_evar y])
  (eval ρ phi2^[evar:dbi↦patt_free_evar y]) =
app_ext (eval ρ phi1^[evar:dbi↦patt_free_evar y])
  (eval ρ phi2^[evar:dbi↦patt_free_evar y])
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
evar_is_fresh_in x phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
well_formed_closed_ex_aux phi2 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
evar_is_fresh_in x phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
well_formed_closed_ex_aux phi1 (S dbi)
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
app_ext (eval ρ phi1^[evar:dbi↦patt_free_evar y])
  (eval ρ phi2^[evar:dbi↦patt_free_evar y]) =
app_ext (eval ρ phi1^[evar:dbi↦patt_free_evar y])
  (eval ρ phi2^[evar:dbi↦patt_free_evar y]) reflexivity.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
size phi2 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
evar_is_fresh_in x phi2 assumption.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
well_formed_closed_ex_aux phi2 (S dbi) now apply andb_true_iff in Hwf.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
size phi1 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
evar_is_fresh_in x phi1 auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux (phi1 $ phi2) (S dbi)
well_formed_closed_ex_aux phi1 (S dbi) now apply andb_true_iff in Hwf.
      + (* Bot. Again, a duplication of the sz=0 case *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
y: evar
Hsz: size ⊥ ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x ⊥
Hwf: well_formed_closed_ex_aux ⊥ (S dbi)
eval ρ ⊥ =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  ⊥^{evar:dbi↦x}
        simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
y: evar
Hsz: size ⊥ ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x ⊥
Hwf: well_formed_closed_ex_aux ⊥ (S dbi)
eval ρ ⊥ =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  ⊥^{evar:dbi↦x} repeat rewrite eval_bott_simpl; auto.
      + (* imp *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 ---> phi2) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 ---> phi2)
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1 ---> phi2)^{evar:dbi↦x}
        mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: size (phi1 ---> phi2) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 ---> phi2)
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} ---> phi2^{evar:dbi↦x})
        simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (phi1 ---> phi2)
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} ---> phi2^{evar:dbi↦x})
        unfold evar_is_fresh_in in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars (phi1 ---> phi2)
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} ---> phi2^{evar:dbi↦x}) simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1 ∪ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} ---> phi2^{evar:dbi↦x}) apply not_elem_of_union in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: (x ∉ free_evars phi1) ∧ x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} ---> phi2^{evar:dbi↦x}) destruct H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
eval ρ
  (phi1^[evar:dbi↦patt_free_evar y] --->
   phi2^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (phi1^{evar:dbi↦x} ---> phi2^{evar:dbi↦x})
        repeat rewrite -> eval_imp_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
⊤ ∖ eval ρ phi1^[evar:dbi↦patt_free_evar y]
∪ eval ρ phi2^[evar:dbi↦patt_free_evar y] =
⊤
∖ eval (update_evar_val x (evar_valuation ρ y) ρ)
    phi1^{evar:dbi↦x}
∪ eval (update_evar_val x (evar_valuation ρ y) ρ)
    phi2^{evar:dbi↦x}
        repeat rewrite <- IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
⊤ ∖ eval ρ phi1^[evar:dbi↦patt_free_evar y]
∪ eval ρ phi2^[evar:dbi↦patt_free_evar y] =
⊤ ∖ eval ρ phi1^[evar:dbi↦patt_free_evar y]
∪ eval ρ phi2^[evar:dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
evar_is_fresh_in x phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
well_formed_closed_ex_aux phi2 (S dbi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
evar_is_fresh_in x phi1
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
well_formed_closed_ex_aux phi1 (S dbi)
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
⊤ ∖ eval ρ phi1^[evar:dbi↦patt_free_evar y]
∪ eval ρ phi2^[evar:dbi↦patt_free_evar y] =
⊤ ∖ eval ρ phi1^[evar:dbi↦patt_free_evar y]
∪ eval ρ phi2^[evar:dbi↦patt_free_evar y] reflexivity.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
size phi2 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
evar_is_fresh_in x phi2 assumption.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
well_formed_closed_ex_aux phi2 (S dbi) now apply andb_true_iff in Hwf.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
size phi1 ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
evar_is_fresh_in x phi1 auto.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi1, phi2: Pattern
y: evar
Hsz: S (size phi1 + size phi2) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi1
H0: x ∉ free_evars phi2
Hwf: well_formed_closed_ex_aux 
  (phi1 ---> phi2) (S dbi)
well_formed_closed_ex_aux phi1 (S dbi) now apply andb_true_iff in Hwf.

      + (* ex *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (ex , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi)^{evar:dbi↦x}
        mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (ex , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi^{evar:S dbi↦x}) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (ex , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi^{evar:S dbi↦x})
        simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi^{evar:S dbi↦x})
        unfold evar_is_fresh_in in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars (ex , phi)
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi^{evar:S dbi↦x}) simpl in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
eval ρ (ex , phi^[evar:S dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (ex , phi^{evar:S dbi↦x})
        repeat rewrite -> eval_ex_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
(let x := fresh_evar phi^[evar:S dbi↦patt_free_evar y]
   in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ'
        phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x})) =
(let x0 := fresh_evar phi^{evar:S dbi↦x} in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x0 e
          (update_evar_val x (evar_valuation ρ y) ρ)
        in
      eval ρ' phi^{evar:S dbi↦x}^{evar:0↦x0})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar
             phi^[evar:S dbi↦patt_free_evar y]) e ρ)
       phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦fresh_evar
                                                 phi^[evar:
                                                 S dbi↦
                                                 patt_free_evar
                                                 y]}) =
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar phi^{evar:S dbi↦x}) e
          (update_evar_val x (evar_valuation ρ y) ρ))
       phi^{evar:S dbi↦x}^{evar:0↦fresh_evar
                                    phi^{evar:S dbi↦x}})
        apply propset_fa_union_same.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
∀ c : M,
  eval
    (update_evar_val
       (fresh_evar phi^[evar:S dbi↦patt_free_evar y])
       c ρ)
    phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦fresh_evar
                                                phi^[evar:
                                                S dbi↦
                                                patt_free_evar
                                                 y]} =
  eval
    (update_evar_val (fresh_evar phi^{evar:S dbi↦x}) c
       (update_evar_val x (evar_valuation ρ y) ρ))
    phi^{evar:S dbi↦x}^{evar:0↦fresh_evar
                                 phi^{evar:S dbi↦x}} intros c.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
eval
  (update_evar_val
     (fresh_evar phi^[evar:S dbi↦patt_free_evar y]) c
     ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦fresh_evar
                                              phi^[evar:
                                              S dbi↦
                                              patt_free_evar
                                                y]} =
eval
  (update_evar_val (fresh_evar phi^{evar:S dbi↦x}) c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi^{evar:S dbi↦x}^{evar:0↦fresh_evar
                               phi^{evar:S dbi↦x}}
        remember (fresh_evar (phi^[evar: S dbi ↦ patt_free_evar y])) as x'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x'} =
eval
  (update_evar_val (fresh_evar phi^{evar:S dbi↦x}) c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi^{evar:S dbi↦x}^{evar:0↦fresh_evar
                               phi^{evar:S dbi↦x}}
        (* rewrite evar_open_bevar_subst_higher.
        { auto. }
        { lia. } *)

        remember (phi^{evar: S dbi ↦ x}) as phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
eval (update_evar_val x' c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x'} =
eval
  (update_evar_val (fresh_evar phi') c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦fresh_evar phi'}
        remember (fresh_evar phi') as Xfr'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
eval (update_evar_val x' c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x'} =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
        remember (evar_fresh (elements (union (free_evars phi') (singleton x')))) as Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
eval (update_evar_val x' c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x'} =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

        (* The same destruct as in the mu case of the set substitution lemma *)
        (* destruct (bevar_occur phi (S dbi)) eqn:Hbevar. *)
        --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
eval (update_evar_val x' c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x'} =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          rewrite evar_open_bevar_subst_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
well_formed_closed_ex_aux (patt_free_evar y) 0 auto. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          remember (union (union (union (union
                                           (union (free_evars phi) (free_evars phi'))
                                           (free_evars (phi^[evar: S dbi ↦ patt_free_evar y])))
                                        (singleton Xfr'))
                                 (singleton x))
                          (singleton y))  as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          remember (evar_fresh (elements B)) as yB.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          assert (HyBx: yB <> x).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
yB ≠ x
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
yB ≠ x solve_fresh_neq. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          assert (HyBXfr': yB <> Xfr').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
yB ≠ Xfr'
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
yB ≠ Xfr' solve_fresh_neq. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          assert (HyBy: yB <> y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
yB ≠ y
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
yB ≠ y solve_fresh_neq. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          assert (HyBfphi: evar_is_fresh_in yB phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
evar_is_fresh_in yB phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
evar_is_fresh_in yB phi
            subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]}
         ∪ {[y]}))) phi
            eapply evar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
evar_fresh
  (elements
     (free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]}
      ∪ {[y]})) ∉ ?B
            2: apply set_evar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi
⊆ free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
  ∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
  ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]} ∪ {[y]}
            rewrite <- 4!union_subseteq_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi
⊆ free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
            apply union_subseteq_l.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          assert (HyBfphi': evar_is_fresh_in yB phi').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
evar_is_fresh_in yB phi'
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
evar_is_fresh_in yB phi'
            subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]}
         ∪ {[y]}))) phi^{evar:S dbi↦x}
            eapply evar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi^{evar:S dbi↦x} ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
evar_fresh
  (elements
     (free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]}
      ∪ {[y]})) ∉ ?B
            2: apply set_evar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi^{evar:S dbi↦x}
⊆ free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
  ∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
  ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]} ∪ {[y]}
            rewrite <- 4!union_subseteq_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi^{evar:S dbi↦x}
⊆ free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
            apply union_subseteq_r.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          assert (HyBfbevar_open: evar_is_fresh_in yB (phi^[evar: S dbi ↦ patt_free_evar y])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
evar_is_fresh_in yB phi^[evar:S dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
evar_is_fresh_in yB phi^[evar:S dbi↦patt_free_evar y] 
            subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi': evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]})))
  phi^{evar:S dbi↦x}
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]}
         ∪ {[y]}))) phi^[evar:S dbi↦patt_free_evar y]
            eapply evar_is_fresh_in_richer'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi': evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]})))
  phi^{evar:S dbi↦x}
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi^[evar:S dbi↦patt_free_evar y] ⊆ ?B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi': evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]})))
  phi^{evar:S dbi↦x}
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
evar_fresh
  (elements
     (free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]}
      ∪ {[y]})) ∉ ?B
            2: apply set_evar_fresh_is_fresh'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi': evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]})))
  phi^{evar:S dbi↦x}
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi^[evar:S dbi↦patt_free_evar y]
⊆ free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
  ∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
  ∪ {[fresh_evar phi^{evar:S dbi↦x}]} ∪ {[x]} ∪ {[y]}
            rewrite <- 3!union_subseteq_l.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfphi': evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]})))
  phi^{evar:S dbi↦x}
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
free_evars phi^[evar:S dbi↦patt_free_evar y]
⊆ free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
  ∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
            apply union_subseteq_r.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val Xfr' c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦Xfr'}

          rewrite -> eval_fresh_evar_open with (x := Xfr') (y := yB).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val yB c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_is_fresh_in Xfr' phi'
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_is_fresh_in yB phi'
          3: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_is_fresh_in yB phi' apply HyBfphi'. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val yB c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_is_fresh_in Xfr' phi'
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_is_fresh_in Xfr' phi' subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
HyBfbevar_open: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars
             phi^{evar:
             S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar
               phi^{evar:
               S dbi↦x}]} ∪ {[x]}
         ∪ {[y]})))
  phi^[evar:S dbi↦patt_free_evar y]
HyBfphi': evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]})))
  phi^{evar:S dbi↦x}
HyBfphi: evar_is_fresh_in
  (evar_fresh
     (elements
        (free_evars phi
         ∪ free_evars phi^{evar:S dbi↦x}
         ∪ free_evars
             phi^[evar:
             S dbi↦patt_free_evar y]
         ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
         ∪ {[x]} ∪ {[y]}))) phi
HyBy: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ y
HyBXfr': evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]}))
≠ fresh_evar phi^{evar:S dbi↦x}
HyBx: evar_fresh
  (elements
     (free_evars phi
      ∪ free_evars phi^{evar:S dbi↦x}
      ∪ free_evars
          phi^[evar:S dbi↦
          patt_free_evar y]
      ∪ {[fresh_evar phi^{evar:S dbi↦x}]}
      ∪ {[x]} ∪ {[y]})) ≠ x
evar_is_fresh_in (fresh_evar phi^{evar:S dbi↦x})
  phi^{evar:S dbi↦x} apply set_evar_fresh_is_fresh. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val yB c
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{evar:0↦yB}
          
          rewrite update_evar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
yB ≠ x
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_evar_val yB c ρ)) phi'^{evar:0↦yB}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
yB ≠ x apply HyBx. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
phi': Pattern
Heqphi': phi' = phi^{evar:S dbi↦x}
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi'
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements (free_evars phi' ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi'
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi'
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_evar_val yB c ρ)) phi'^{evar:0↦yB}
          subst phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_evar_val yB c ρ))
  phi^{evar:S dbi↦x}^{evar:0↦yB}
          rewrite evar_open_comm_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_evar_val yB c ρ))
  phi^{evar:0↦yB}^{evar:
  Init.Nat.pred (S dbi)↦x}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_evar_val yB c ρ))
  phi^{evar:0↦yB}^{evar:Init.Nat.pred (S dbi)↦x}

          remember (update_evar_val yB c ρ) as ρ'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval (update_evar_val x (evar_valuation ρ y) ρ')
  phi^{evar:0↦yB}^{evar:Init.Nat.pred (S dbi)↦x}
          remember (phi^{evar: 0 ↦ yB}^{evar: S dbi ↦ x}) as phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
phi': Pattern
Heqphi': phi' = phi^{evar:0↦yB}^{evar:S dbi↦x}
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval (update_evar_val x (evar_valuation ρ y) ρ')
  phi^{evar:0↦yB}^{evar:Init.Nat.pred (S dbi)↦x}

          subst phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval (update_evar_val x (evar_valuation ρ y) ρ')
  phi^{evar:0↦yB}^{evar:Init.Nat.pred (S dbi)↦x}

          assert (Hevar_val: evar_valuation ρ y = evar_valuation ρ' y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
evar_valuation ρ y = evar_valuation ρ' y
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:
  Init.Nat.pred (S dbi)↦
  patt_free_evar y] =
eval (update_evar_val x (evar_valuation ρ y) ρ')
  phi^{evar:0↦yB}^{evar:
  Init.Nat.pred (S dbi)↦x}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
evar_valuation ρ y = evar_valuation ρ' y subst ρ'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y rewrite update_evar_val_neq.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
yB ≠ y
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_valuation ρ y = evar_valuation ρ y apply HyBy.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
evar_valuation ρ y = evar_valuation ρ y reflexivity. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval (update_evar_val x (evar_valuation ρ y) ρ')
  phi^{evar:0↦yB}^{evar:Init.Nat.pred (S dbi)↦x}
          rewrite Hevar_val.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval (update_evar_val x (evar_valuation ρ' y) ρ')
  phi^{evar:0↦yB}^{evar:Init.Nat.pred (S dbi)↦x}

          rewrite <- IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval ρ'
  phi^{evar:0↦yB}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          subst ρ'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
eval (update_evar_val x' c ρ)
  phi^{evar:0↦x'}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y] =
eval (update_evar_val yB c ρ)
  phi^{evar:0↦yB}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          rewrite <- evar_open_bevar_subst_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
eval (update_evar_val x' c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x'} =
eval (update_evar_val yB c ρ)
  phi^{evar:0↦yB}^[evar:Init.Nat.pred (S dbi)↦patt_free_evar
                                                y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          rewrite <- evar_open_bevar_subst_higher.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
eval (update_evar_val x' c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦x'} =
eval (update_evar_val yB c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          rewrite -> eval_fresh_evar_open with (y := yB).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
eval (update_evar_val yB c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦yB} =
eval (update_evar_val yB c ρ)
  phi^[evar:S dbi↦patt_free_evar y]^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in x' phi^[evar:S dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in yB phi^[evar:S dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in x' phi^[evar:S dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in yB phi^[evar:S dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))

          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in x' phi^[evar:S dbi↦patt_free_evar y] rewrite Heqx'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in
  (fresh_evar phi^[evar:S dbi↦patt_free_evar y])
  phi^[evar:S dbi↦patt_free_evar y] apply set_evar_fresh_is_fresh. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in yB phi^[evar:S dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
evar_is_fresh_in yB phi^[evar:S dbi↦patt_free_evar y] apply HyBfbevar_open. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0 auto. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
well_formed_closed_ex_aux (patt_free_evar y) 0 auto. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
Hevar_val: evar_valuation ρ y =
evar_valuation (update_evar_val yB c ρ) y
0 < S dbi lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi^{evar:0↦yB} ≤ sz rewrite <- evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
size phi ≤ sz lia. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi^{evar:0↦yB} apply evar_is_fresh_in_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
x ≠ yB
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi apply not_eq_sym.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
yB ≠ x
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi apply HyBx.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
evar_is_fresh_in x phi apply H. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi)))
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB}
  (S (Init.Nat.pred (S dbi))) replace (pred (S dbi)) with dbi by lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
well_formed_closed_ex_aux phi^{evar:0↦yB} (S dbi) apply wfc_mu_aux_body_ex_imp3; auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: x ∉ free_evars phi
Hwf: well_formed_closed_ex_aux (ex , phi) (S dbi)
c: M
x': evar
Heqx': x' =
fresh_evar phi^[evar:S dbi↦patt_free_evar y]
Xfr': evar
HeqXfr': Xfr' = fresh_evar phi^{evar:S dbi↦x}
Xu: evar
HeqXu: Xu =
evar_fresh
  (elements
     (free_evars phi^{evar:S dbi↦x} ∪ {[x']}))
B: EVarSet
HeqB: B =
free_evars phi ∪ free_evars phi^{evar:S dbi↦x}
∪ free_evars phi^[evar:S dbi↦patt_free_evar y]
∪ {[Xfr']} ∪ {[x]} ∪ {[y]}
yB: evar
HeqyB: yB = evar_fresh (elements B)
HyBx: yB ≠ x
HyBXfr': yB ≠ Xfr'
HyBy: yB ≠ y
HyBfphi: evar_is_fresh_in yB phi
HyBfphi': evar_is_fresh_in yB phi^{evar:S dbi↦x}
HyBfbevar_open: evar_is_fresh_in yB
  phi^[evar:S dbi↦patt_free_evar y]
ρ': Valuation
Heqρ': ρ' = update_evar_val yB c ρ
Hevar_val: evar_valuation ρ y = evar_valuation ρ' y
0 ≤ S dbi lia. }
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
eval ρ (mu , phi^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (mu , phi)^{evar:dbi↦x} mlSimpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
eval ρ (mu , phi^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (mu , phi^{evar:increase_mu pm_spec_data dbi↦x}) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
eval ρ (mu , phi^[evar:dbi↦patt_free_evar y]) =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  (mu , phi^{evar:dbi↦x})
        rewrite 2!eval_mu_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
(let X := fresh_svar phi^[evar:dbi↦patt_free_evar y]
   in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ'
        phi^[evar:dbi↦patt_free_evar y]^{svar:0↦X})) =
(let X := fresh_svar phi^{evar:dbi↦x} in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' :=
        update_svar_val X S
          (update_evar_val x (evar_valuation ρ y) ρ)
        in
      eval ρ' phi^{evar:dbi↦x}^{svar:0↦X})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[evar:dbi↦patt_free_evar y])
          S ρ)
       phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                                 phi^[evar:dbi↦
                                                 patt_free_evar
                                                 y]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi^{evar:dbi↦x})
          S (update_evar_val x (evar_valuation ρ y) ρ))
       phi^{evar:dbi↦x}^{svar:0↦fresh_svar
                                  phi^{evar:dbi↦x}})
        apply f_equal.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
(λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[evar:dbi↦patt_free_evar y]) S
        ρ)
     phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                               phi^[evar:dbi↦
                                               patt_free_evar
                                                 y]}) =
(λ S : propset M,
   eval
     (update_svar_val (fresh_svar phi^{evar:dbi↦x}) S
        (update_evar_val x (evar_valuation ρ y) ρ))
     phi^{evar:dbi↦x}^{svar:0↦fresh_svar
                                phi^{evar:dbi↦x}}) apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
∀ x0 : propset M,
  eval
    (update_svar_val
       (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0
       ρ)
    phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                              phi^[evar:dbi↦
                                              patt_free_evar
                                                y]} =
  eval
    (update_svar_val (fresh_svar phi^{evar:dbi↦x}) x0
       (update_evar_val x (evar_valuation ρ y) ρ))
    phi^{evar:dbi↦x}^{svar:0↦fresh_svar
                               phi^{evar:dbi↦x}} intros x0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval
  (update_svar_val (fresh_svar phi^{evar:dbi↦x}) x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi^{evar:dbi↦x}^{svar:0↦fresh_svar phi^{evar:dbi↦x}}
        (* fold (evar_open dbi y phi). *)
        remember (phi^{evar: dbi ↦ x}) as phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval
  (update_svar_val (fresh_svar phi') x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦fresh_svar phi'}
        remember (fresh_svar phi') as Xfr'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval
  (update_svar_val Xfr' x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xfr'}
        remember (svar_fresh (elements (free_svars phi'))) as Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval
  (update_svar_val Xfr' x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xfr'}
        remember (update_evar_val x (evar_valuation ρ y) ρ) as ρ'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval (update_svar_val Xfr' x0 ρ') phi'^{svar:0↦Xfr'}
        pose proof (Hfresh_subst := eval_fresh_svar_open phi' Xfr' Xu x0 0 ρ').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval (update_svar_val Xfr' x0 ρ') phi'^{svar:0↦Xfr'}
        rewrite -> Hfresh_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
svar_is_fresh_in Xfr' phi'
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
svar_is_fresh_in Xu phi'
        2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
svar_is_fresh_in Xfr' phi' subst Xfr'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in (fresh_svar phi') phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val
         (fresh_svar phi') x0 ρ')
      phi'^{svar:0↦fresh_svar phi'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
svar_is_fresh_in (fresh_svar phi') phi' apply set_svar_fresh_is_fresh. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
svar_is_fresh_in Xu phi'
        2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
svar_is_fresh_in Xu phi' subst Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in
    (svar_fresh
       (elements (free_svars phi')))
    phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements (free_svars phi')))
         x0 ρ')
      phi'^{svar:0↦svar_fresh
                     (elements
                     (free_svars phi'))}
svar_is_fresh_in
  (svar_fresh (elements (free_svars phi'))) phi' apply set_svar_fresh_is_fresh'. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
eval
  (update_svar_val
     (fresh_svar phi^[evar:dbi↦patt_free_evar y]) x0 ρ)
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}

        remember (update_svar_val (fresh_svar (phi^[evar: dbi ↦ patt_free_evar y])) x0 ρ) as ρs1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
eval ρs1'
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
        remember (update_svar_val Xu x0 ρ) as ρs2'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
eval ρs1'
  phi^[evar:dbi↦patt_free_evar y]^{svar:0↦fresh_svar
                                            phi^[evar:dbi↦
                                            patt_free_evar
                                              y]} =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
        rewrite -> svar_open_bevar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
eval ρs1'
  phi^{svar:0↦fresh_svar
                phi^[evar:dbi↦patt_free_evar y]}^[evar:dbi↦
  patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
well_formed_closed (patt_free_evar y) 2: wf_auto2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
eval ρs1'
  phi^{svar:0↦fresh_svar
                phi^[evar:dbi↦patt_free_evar y]}^[evar:dbi↦
  patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
        remember (fresh_svar (phi^[evar: dbi ↦ patt_free_evar y])) as Xfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}

        (* dbi may or may not occur in phi1 *)
        remember (bevar_occur phi dbi) as Hoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
Hoc: bool
HeqHoc: Hoc = bevar_occur phi dbi
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
        move: HeqHoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
Hoc: bool
Hoc = bevar_occur phi dbi
→ eval ρs1'
    phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
  eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
        case: Hoc => HeqHoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: true = bevar_occur phi dbi
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: false = bevar_occur phi dbi
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦
  patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
        -- (* dbi ocurs in phi1 *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: true = bevar_occur phi dbi
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          pose proof (HXfr1Fresh := set_svar_fresh_is_fresh (phi^[evar: dbi ↦ patt_free_evar y])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: true = bevar_occur phi dbi
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          rewrite <- HeqXfr1 in HXfr1Fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: true = bevar_occur phi dbi
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          symmetry in HeqHoc.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          pose proof (Hsub := bevar_subst_contains_subformula phi (patt_free_evar y) dbi HeqHoc).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          pose proof (HXfr1Fresh2 := svar_fresh_in_subformula _ _ _ Hsub HXfr1Fresh).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xfr1 (patt_free_evar y)
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}

          assert (HXu: (Xfr1 = Xu)).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xfr1 (patt_free_evar y)
Xfr1 = Xu
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xfr1 (patt_free_evar y)
HXu: Xfr1 = Xu
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦
  patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xfr1 (patt_free_evar y)
Xfr1 = Xu subst Xfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
fresh_svar phi^[evar:dbi↦patt_free_evar y] = Xu subst Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in
    (svar_fresh
       (elements (free_svars phi')))
    phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements (free_svars phi')))
         x0 ρ')
      phi'^{svar:0↦svar_fresh
                     (elements
                     (free_svars phi'))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh (elements (free_svars phi')))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
fresh_svar phi^[evar:dbi↦patt_free_evar y] =
svar_fresh (elements (free_svars phi')) subst phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
fresh_svar phi^[evar:dbi↦patt_free_evar y] =
svar_fresh (elements (free_svars phi^{evar:dbi↦x})) unfold fresh_svar.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
svar_fresh_s
  (free_svars phi^[evar:dbi↦patt_free_evar y]) =
svar_fresh (elements (free_svars phi^{evar:dbi↦x}))
            rewrite -> free_svars_bevar_subst_eq.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
svar_fresh_s
  (free_svars phi ∪ free_svars (patt_free_evar y)) =
svar_fresh (elements (free_svars phi^{evar:dbi↦x}))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
bevar_occur phi dbi
            rewrite -> free_svars_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
svar_fresh_s
  (free_svars phi ∪ free_svars (patt_free_evar y)) =
svar_fresh (elements (free_svars phi))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
bevar_occur phi dbi
            assert (Hsvev: free_svars (patt_free_evar y) = ∅) by auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
Hsvev: free_svars (patt_free_evar y) = ∅
svar_fresh_s
  (free_svars phi ∪ free_svars (patt_free_evar y)) =
svar_fresh (elements (free_svars phi))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
bevar_occur phi dbi
            rewrite Hsvev.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
Hsvev: free_svars (patt_free_evar y) = ∅
svar_fresh_s (free_svars phi ∪ ∅) =
svar_fresh (elements (free_svars phi))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
bevar_occur phi dbi
            rewrite union_empty_r_L.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
Hsvev: free_svars (patt_free_evar y) = ∅
svar_fresh_s (free_svars phi) =
svar_fresh (elements (free_svars phi))
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
bevar_occur phi dbi
            reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval (update_svar_val Xfr' x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0 ρ')
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
ρs1': Valuation
Heqρs1': ρs1' =
update_svar_val
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y]) x0 ρ
ρs2': Valuation
Heqρs2': ρs2' =
update_svar_val
  (svar_fresh
     (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (fresh_svar
     phi^[evar:dbi↦patt_free_evar y])
  (patt_free_evar y)
bevar_occur phi dbi
            assumption.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xfr1
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xfr1 (patt_free_evar y)
HXu: Xfr1 = Xu
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}

          rewrite -> HXu in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
eval ρs1' phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          replace (ρs1') with (ρs2').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
eval ρs2' phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
ρs2' = ρs1'
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
ρs2' = ρs1' subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Hfresh_subst: svar_is_fresh_in
  (fresh_svar phi^{evar:dbi↦x})
  phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval
      (update_svar_val
         (fresh_svar phi^{evar:dbi↦x})
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      fresh_svar phi^{evar:dbi↦x}} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
HeqXfr1: svar_fresh
  (elements (free_svars phi^{evar:dbi↦x})) =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi^{evar:dbi↦x})))
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi^{evar:dbi↦x})))
  (patt_free_evar y)
update_svar_val
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ =
update_svar_val
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ reflexivity. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
eval ρs2' phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
          subst ρs2' ρ'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xu}
          assert (Hs1s1': update_svar_val Xu x0 ρs1' = update_svar_val Xu x0 ρ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
update_svar_val Xu x0 ρs1' = update_svar_val Xu x0 ρ
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
Hs1s1': update_svar_val Xu x0 ρs1' =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦
  patt_free_evar y] =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xu}
          {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
update_svar_val Xu x0 ρs1' = update_svar_val Xu x0 ρ subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Hfresh_subst: svar_is_fresh_in
  (fresh_svar phi^{evar:dbi↦x})
  phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval
      (update_svar_val
         (fresh_svar phi^{evar:dbi↦x})
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      fresh_svar phi^{evar:dbi↦x}} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
HeqXfr1: svar_fresh
  (elements (free_svars phi^{evar:dbi↦x})) =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi^{evar:dbi↦x})))
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi^{evar:dbi↦x})))
  (patt_free_evar y)
update_svar_val
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  x0
  (update_svar_val
     (svar_fresh
        (elements (free_svars phi^{evar:dbi↦x}))) x0 ρ) =
update_svar_val
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ rewrite update_svar_val_shadow.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Hfresh_subst: svar_is_fresh_in
  (fresh_svar phi^{evar:dbi↦x})
  phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval
      (update_svar_val
         (fresh_svar phi^{evar:dbi↦x})
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      fresh_svar phi^{evar:dbi↦x}} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
HeqXfr1: svar_fresh
  (elements (free_svars phi^{evar:dbi↦x})) =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi^{evar:dbi↦x})))
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in
  (svar_fresh
     (elements
        (free_svars phi^{evar:dbi↦x})))
  (patt_free_evar y)
update_svar_val
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ =
update_svar_val
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  x0 ρ reflexivity. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
Hs1s1': update_svar_val Xu x0 ρs1' =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xu}

          subst phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
Hs1s1': update_svar_val Xu x0 ρs1' =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi^{evar:dbi↦x}^{svar:0↦Xu}
          rewrite <- svar_open_evar_open_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
Hs1s1': update_svar_val Xu x0 ρs1' =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi^{svar:0↦Xu}^{evar:dbi↦x}
          rewrite -update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
HXu: Xfr1 = Xu
Hs1s1': update_svar_val Xu x0 ρs1' =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_svar_val Xu x0 ρ))
  phi^{svar:0↦Xu}^{evar:dbi↦x} subst Xfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
ρs1': Valuation
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xu x0 ρ
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0 ρs1' =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_svar_val Xu x0 ρ))
  phi^{svar:0↦Xu}^{evar:dbi↦x}
          subst ρs1'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_svar_val Xu x0 ρ))
  phi^{svar:0↦Xu}^{evar:dbi↦x}
          replace (evar_valuation ρ y) with (evar_valuation (update_svar_val Xu x0 ρ) y).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_evar_val x
     (evar_valuation (update_svar_val Xu x0 ρ) y)
     (update_svar_val Xu x0 ρ))
  phi^{svar:0↦Xu}^{evar:dbi↦x}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
evar_valuation (update_svar_val Xu x0 ρ) y =
evar_valuation ρ y
          2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
evar_valuation (update_svar_val Xu x0 ρ) y =
evar_valuation ρ y
            destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation
               {|
                 evar_valuation := ρₑ;
                 svar_valuation := ρₛ
               |} y)
            {|
              evar_valuation := ρₑ;
              svar_valuation := ρₛ
            |}))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation
               {|
                 evar_valuation := ρₑ;
                 svar_valuation := ρₛ
               |} y)
            {|
              evar_valuation := ρₑ;
              svar_valuation := ρₛ
            |}))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0
     {|
       evar_valuation := ρₑ;
       svar_valuation := ρₛ
     |}) =
update_svar_val Xu x0
  {|
    evar_valuation := ρₑ;
    svar_valuation := ρₛ
  |}
evar_valuation
  (update_svar_val Xu x0
     {| evar_valuation := ρₑ; svar_valuation := ρₛ |})
  y =
evar_valuation
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} y simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρₑ: evar → M
ρₛ: svar → propset M
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation
               {|
                 evar_valuation := ρₑ;
                 svar_valuation := ρₛ
               |} y)
            {|
              evar_valuation := ρₑ;
              svar_valuation := ρₛ
            |}))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation
               {|
                 evar_valuation := ρₑ;
                 svar_valuation := ρₛ
               |} y)
            {|
              evar_valuation := ρₑ;
              svar_valuation := ρₛ
            |}))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0
     {|
       evar_valuation := ρₑ;
       svar_valuation := ρₛ
     |}) =
update_svar_val Xu x0
  {|
    evar_valuation := ρₑ;
    svar_valuation := ρₛ
  |}
ρₑ y = ρₑ y reflexivity.
          }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval
  (update_evar_val x
     (evar_valuation (update_svar_val Xu x0 ρ) y)
     (update_svar_val Xu x0 ρ))
  phi^{svar:0↦Xu}^{evar:dbi↦x}

          rewrite <- IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
size phi^{svar:0↦Xu} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
evar_is_fresh_in x phi^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
well_formed_closed_ex_aux phi^{svar:0↦Xu} (S dbi)
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ)
  phi^{svar:0↦Xu}^[evar:dbi↦patt_free_evar y] reflexivity.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
size phi^{svar:0↦Xu} ≤ sz rewrite <- svar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
size phi ≤ sz simpl in Hsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: S (size phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
size phi ≤ sz lia.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
evar_is_fresh_in x phi^{svar:0↦Xu} apply evar_fresh_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
evar_is_fresh_in x phi apply evar_is_fresh_in_mu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
evar_is_fresh_in x phi apply H.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
well_formed_closed_ex_aux phi^{svar:0↦Xu} (S dbi) apply wfc_ex_aux_body_mu_imp1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
well_formed_closed_ex_aux phi (S dbi) = true eapply well_formed_closed_ex_aux_ind.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
?ind_evar1 ≤ S dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
well_formed_closed_ex_aux phi ?ind_evar1 = true
          2: exact Hwf.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqXfr1: Xu =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc: bevar_occur phi dbi = true
HXfr1Fresh: svar_is_fresh_in Xu
  phi^[evar:dbi↦patt_free_evar y]
Hsub: is_subformula_of_ind (patt_free_evar y)
  phi^[evar:dbi↦patt_free_evar y]
HXfr1Fresh2: svar_is_fresh_in Xu (patt_free_evar y)
Hs1s1': update_svar_val Xu x0
  (update_svar_val Xu x0 ρ) =
update_svar_val Xu x0 ρ
S dbi ≤ S dbi auto.

          -- (* dbi does not occur in phi1 *)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc: false = bevar_occur phi dbi
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
            pose proof (HeqHoc' := HeqHoc).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval ρs1'
  phi^{svar:0↦Xfr1}^[evar:dbi↦patt_free_evar y] =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
            rewrite -> bevar_subst_not_occur.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval ρs1' phi^{svar:0↦Xfr1} =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc, HeqHoc': false = bevar_occur phi dbi
well_formed_closed_ex_aux phi^{svar:0↦Xfr1} dbi
            2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc, HeqHoc': false = bevar_occur phi dbi
well_formed_closed_ex_aux phi^{svar:0↦Xfr1} dbi apply wfc_ex_aux_body_mu_imp1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc, HeqHoc': false = bevar_occur phi dbi
well_formed_closed_ex_aux phi dbi = true
              apply wfc_ex_lower; wf_auto2.
            }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
ρ': Valuation
Heqρ': ρ' = update_evar_val x (evar_valuation ρ y) ρ
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval (update_svar_val Xfr' x0 ρ')
      phi'^{svar:0↦Xfr'} =
    eval (update_svar_val Xu x0 ρ')
      phi'^{svar:0↦Xu}
ρs1': Valuation
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
Heqρs1': ρs1' = update_svar_val Xfr1 x0 ρ
ρs2': Valuation
Heqρs2': ρs2' = update_svar_val Xu x0 ρ
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval ρs1' phi^{svar:0↦Xfr1} =
eval (update_svar_val Xu x0 ρ') phi'^{svar:0↦Xu}
            (* Now svar_open does nothing to phi1, since it does not contain dbi (see HeqHoc). *)
            (* symmetry in HeqHoc. apply evar_open_not_occur with (x1:=x) in HeqHoc. *)
            (* X is not free in phi1, so the fact that in evar_val' it is updated to some 
           value is irrelevant. *)

            subst ρs1' ρs2' ρ'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xfr1 x0 ρ) phi^{svar:0↦Xfr1} =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xu}
            replace (eval (update_svar_val Xu x0 (update_evar_val x (evar_valuation ρ y) ρ)) (phi'^{svar: 0 ↦ Xu}))
            with (eval (update_svar_val Xu x0 ρ) (phi'^{svar: 0 ↦ Xu})).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xfr1 x0 ρ) phi^{svar:0↦Xfr1} =
eval (update_svar_val Xu x0 ρ) phi'^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xu x0 ρ) phi'^{svar:0↦Xu} =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xu}
            2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xu x0 ρ) phi'^{svar:0↦Xu} =
eval
  (update_svar_val Xu x0
     (update_evar_val x (evar_valuation ρ y) ρ))
  phi'^{svar:0↦Xu}
              rewrite -update_evar_val_svar_val_comm.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xu x0 ρ) phi'^{svar:0↦Xu} =
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_svar_val Xu x0 ρ)) phi'^{svar:0↦Xu}
              symmetry.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval
  (update_evar_val x (evar_valuation ρ y)
     (update_svar_val Xu x0 ρ)) phi'^{svar:0↦Xu} =
eval (update_svar_val Xu x0 ρ) phi'^{svar:0↦Xu}
              apply eval_free_evar_independent.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
evar_is_fresh_in x phi'^{svar:0↦Xu}
              unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
x ∉ free_evars phi'^{svar:0↦Xu}
              rewrite -> free_evars_svar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
x ∉ free_evars phi' subst phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
x ∉ free_evars phi^{evar:dbi↦x}
              rewrite evar_open_not_occur; auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
well_formed_closed_ex_aux phi dbi
              simpl in Hwf.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux phi (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
well_formed_closed_ex_aux phi dbi apply wfc_ex_lower; auto.
            }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
phi': Pattern
Heqphi': phi' = phi^{evar:dbi↦x}
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi'
Xu: svar
HeqXu: Xu = svar_fresh (elements (free_svars phi'))
Hfresh_subst: svar_is_fresh_in Xfr' phi'
→ svar_is_fresh_in Xu phi'
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi'^{svar:0↦Xu}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xfr1 x0 ρ) phi^{svar:0↦Xfr1} =
eval (update_svar_val Xu x0 ρ) phi'^{svar:0↦Xu}
            subst phi'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xfr1 x0 ρ) phi^{svar:0↦Xfr1} =
eval (update_svar_val Xu x0 ρ)
  phi^{evar:dbi↦x}^{svar:0↦Xu} rewrite evar_open_not_occur; auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
well_formed_closed_ex_aux phi dbi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xfr1 x0 ρ) phi^{svar:0↦Xfr1} =
eval (update_svar_val Xu x0 ρ) phi^{svar:0↦Xu}
            {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
well_formed_closed_ex_aux phi dbi apply wfc_ex_lower; auto. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xfr1 x0 ρ) phi^{svar:0↦Xfr1} =
eval (update_svar_val Xu x0 ρ) phi^{svar:0↦Xu}
            rewrite -> eval_fresh_svar_open with (Y := Xu).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
eval (update_svar_val Xu x0 ρ) phi^{svar:0↦Xu} =
eval (update_svar_val Xu x0 ρ) phi^{svar:0↦Xu}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in Xfr1 phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in Xu phi reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in Xfr1 phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in Xu phi
            {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in Xfr1 phi subst Xfr1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in
  (fresh_svar phi^[evar:dbi↦patt_free_evar y]) phi
              simpl in Hwf.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux phi (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in
  (fresh_svar phi^[evar:dbi↦patt_free_evar y]) phi symmetry in HeqHoc'.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux phi (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqHoc: false = bevar_occur phi dbi
HeqHoc': bevar_occur phi dbi = false
svar_is_fresh_in
  (fresh_svar phi^[evar:dbi↦patt_free_evar y]) phi
              apply wfc_ex_lower in Hwf; auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux phi dbi = true
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqHoc: false = bevar_occur phi dbi
HeqHoc': bevar_occur phi dbi = false
svar_is_fresh_in
  (fresh_svar phi^[evar:dbi↦patt_free_evar y]) phi
              pose proof (Hsubst := bevar_subst_not_occur _ (patt_free_evar y) _ Hwf).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux phi dbi = true
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
HeqHoc: false = bevar_occur phi dbi
HeqHoc': bevar_occur phi dbi = false
Hsubst: phi^[evar:dbi↦patt_free_evar y] = phi
svar_is_fresh_in
  (fresh_svar phi^[evar:dbi↦patt_free_evar y]) phi
              rewrite Hsubst; auto.
            }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in Xu phi
            {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
svar_is_fresh_in Xu phi pose proof (Hfr := set_svar_fresh_is_fresh').
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
Hfr: ∀ S : SVarSet, svar_fresh (elements S) ∉ S
svar_is_fresh_in Xu phi
              specialize (Hfr (free_svars (phi^{evar: dbi ↦ x}))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Xu: svar
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in Xu phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val Xu x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xu}
HeqXu: Xu =
svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
Hfr: svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
∉ free_svars phi^{evar:dbi↦x}
svar_is_fresh_in Xu phi
              subst Xu.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
Hfr: svar_fresh
  (elements (free_svars phi^{evar:dbi↦x}))
∉ free_svars phi^{evar:dbi↦x}
svar_is_fresh_in
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  phi
              rewrite free_svars_evar_open in Hfr.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
Hfr: svar_fresh (elements (free_svars phi))
∉ free_svars phi
svar_is_fresh_in
  (svar_fresh (elements (free_svars phi^{evar:dbi↦x})))
  phi
              rewrite free_svars_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (dbi : db_index) (phi : Pattern) (y : evar),
  size phi ≤ sz
  → ∀ (ρ : Valuation) (x : evar),
      evar_is_fresh_in x phi
      → well_formed_closed_ex_aux phi (S dbi)
        → eval ρ
            phi^[evar:dbi↦
            patt_free_evar y] =
          eval
            (update_evar_val x
               (evar_valuation ρ y) ρ)
            phi^{evar:dbi↦x}
dbi: db_index
phi: Pattern
y: evar
Hsz: size (mu , phi) ≤ S sz
ρ: Valuation
x: evar
H: evar_is_fresh_in x (mu , phi)
Hwf: well_formed_closed_ex_aux (mu , phi) (S dbi)
x0: propset M
Xfr': svar
HeqXfr': Xfr' = fresh_svar phi^{evar:dbi↦x}
Hfresh_subst: svar_is_fresh_in Xfr' phi^{evar:dbi↦x}
→ svar_is_fresh_in
    (svar_fresh
       (elements
          (free_svars phi^{evar:dbi↦x})))
    phi^{evar:dbi↦x}
  → eval
      (update_svar_val Xfr' x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦Xfr'} =
    eval
      (update_svar_val
         (svar_fresh
            (elements
               (free_svars
                  phi^{evar:dbi↦x})))
         x0
         (update_evar_val x
            (evar_valuation ρ y) ρ))
      phi^{evar:dbi↦x}^{svar:0↦
      svar_fresh
        (elements
           (free_svars phi^{evar:dbi↦x}))}
Xfr1: svar
HeqXfr1: Xfr1 =
fresh_svar phi^[evar:dbi↦patt_free_evar y]
HeqHoc, HeqHoc': false = bevar_occur phi dbi
Hfr: svar_fresh (elements (free_svars phi))
∉ free_svars phi
svar_is_fresh_in
  (svar_fresh (elements (free_svars phi))) phi auto.
            }
  Qed.

  Lemma element_substitution_lemma
        (phi : Pattern) (x y : evar) (ρ : Valuation) (dbi : db_index) :
    evar_is_fresh_in x phi -> well_formed_closed_ex_aux phi (S dbi) ->
    eval ρ (phi^[evar: dbi ↦ patt_free_evar y])
    = @eval M (update_evar_val x (evar_valuation ρ y) ρ) (phi^{evar: dbi ↦ x}).
signature: Signature
M: Model
phi: Pattern
x, y: evar
ρ: Valuation
dbi: db_index
evar_is_fresh_in x phi
→ well_formed_closed_ex_aux phi (S dbi)
  → eval ρ phi^[evar:dbi↦patt_free_evar y] =
    eval (update_evar_val x (evar_valuation ρ y) ρ)
      phi^{evar:dbi↦x}
  Proof.
signature: Signature
M: Model
phi: Pattern
x, y: evar
ρ: Valuation
dbi: db_index
evar_is_fresh_in x phi
→ well_formed_closed_ex_aux phi (S dbi)
  → eval ρ phi^[evar:dbi↦patt_free_evar y] =
    eval (update_evar_val x (evar_valuation ρ y) ρ)
      phi^{evar:dbi↦x}
    intros.
signature: Signature
M: Model
phi: Pattern
x, y: evar
ρ: Valuation
dbi: db_index
H: evar_is_fresh_in x phi
H0: well_formed_closed_ex_aux phi (S dbi)
eval ρ phi^[evar:dbi↦patt_free_evar y] =
eval (update_evar_val x (evar_valuation ρ y) ρ)
  phi^{evar:dbi↦x}
    apply Private_plugging_patterns_bevar_subst with (sz := size phi);auto.
  Qed.

  (* eval unchanged within subformula over fresh element variable *)
  Lemma eval_fresh_evar_subterm ϕ₁ ϕ₂ c dbi ρ :
    is_subformula_of_ind ϕ₁ ϕ₂ ->
    @eval M (update_evar_val (fresh_evar ϕ₂) c ρ) (ϕ₁^{evar: dbi ↦ (fresh_evar ϕ₂)})
    = eval (update_evar_val (fresh_evar ϕ₁) c ρ) (ϕ₁^{evar: dbi ↦ (fresh_evar ϕ₁)}).
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
c: M
dbi: db_index
ρ: Valuation
is_subformula_of_ind ϕ₁ ϕ₂
→ eval (update_evar_val (fresh_evar ϕ₂) c ρ)
    ϕ₁^{evar:dbi↦fresh_evar ϕ₂} =
  eval (update_evar_val (fresh_evar ϕ₁) c ρ)
    ϕ₁^{evar:dbi↦fresh_evar ϕ₁}
  Proof.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
c: M
dbi: db_index
ρ: Valuation
is_subformula_of_ind ϕ₁ ϕ₂
→ eval (update_evar_val (fresh_evar ϕ₂) c ρ)
    ϕ₁^{evar:dbi↦fresh_evar ϕ₂} =
  eval (update_evar_val (fresh_evar ϕ₁) c ρ)
    ϕ₁^{evar:dbi↦fresh_evar ϕ₁}
    intros Hsub.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
c: M
dbi: db_index
ρ: Valuation
Hsub: is_subformula_of_ind ϕ₁ ϕ₂
eval (update_evar_val (fresh_evar ϕ₂) c ρ)
  ϕ₁^{evar:dbi↦fresh_evar ϕ₂} =
eval (update_evar_val (fresh_evar ϕ₁) c ρ)
  ϕ₁^{evar:dbi↦fresh_evar ϕ₁}
    apply eval_fresh_evar_open; auto.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
c: M
dbi: db_index
ρ: Valuation
Hsub: is_subformula_of_ind ϕ₁ ϕ₂
evar_is_fresh_in (fresh_evar ϕ₂) ϕ₁
    eapply evar_fresh_in_subformula.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
c: M
dbi: db_index
ρ: Valuation
Hsub: is_subformula_of_ind ϕ₁ ϕ₂
is_subformula_of_ind ϕ₁ ?ϕ₂
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
c: M
dbi: db_index
ρ: Valuation
Hsub: is_subformula_of_ind ϕ₁ ϕ₂
evar_is_fresh_in (fresh_evar ϕ₂) ?ϕ₂ apply Hsub.
signature: Signature
M: Model
ϕ₁, ϕ₂: Pattern
c: M
dbi: db_index
ρ: Valuation
Hsub: is_subformula_of_ind ϕ₁ ϕ₂
evar_is_fresh_in (fresh_evar ϕ₂) ϕ₂
    apply set_evar_fresh_is_fresh.
  Qed.

  (* model predicate of evar_open is maintained with change of variables *)
  Lemma M_predicate_evar_open_fresh_evar_1 x₁ x₂ ϕ :
    evar_is_fresh_in x₁ ϕ ->
    evar_is_fresh_in x₂ ϕ ->
    M_predicate M (ϕ^{evar: 0 ↦ x₁}) ->
    M_predicate M (ϕ^{evar: 0 ↦ x₂}).
signature: Signature
M: Model
x₁, x₂: evar
ϕ: Pattern
evar_is_fresh_in x₁ ϕ
→ evar_is_fresh_in x₂ ϕ
  → M_predicate M ϕ^{evar:0↦x₁}
    → M_predicate M ϕ^{evar:0↦x₂}
  Proof.
signature: Signature
M: Model
x₁, x₂: evar
ϕ: Pattern
evar_is_fresh_in x₁ ϕ
→ evar_is_fresh_in x₂ ϕ
  → M_predicate M ϕ^{evar:0↦x₁}
    → M_predicate M ϕ^{evar:0↦x₂}
    intros Hfr1 Hfr2.
signature: Signature
M: Model
x₁, x₂: evar
ϕ: Pattern
Hfr1: evar_is_fresh_in x₁ ϕ
Hfr2: evar_is_fresh_in x₂ ϕ
M_predicate M ϕ^{evar:0↦x₁}
→ M_predicate M ϕ^{evar:0↦x₂}
    unfold evar_is_fresh_in in *.
signature: Signature
M: Model
x₁, x₂: evar
ϕ: Pattern
Hfr1: x₁ ∉ free_evars ϕ
Hfr2: x₂ ∉ free_evars ϕ
M_predicate M ϕ^{evar:0↦x₁}
→ M_predicate M ϕ^{evar:0↦x₂}
    unfold M_predicate.
signature: Signature
M: Model
x₁, x₂: evar
ϕ: Pattern
Hfr1: x₁ ∉ free_evars ϕ
Hfr2: x₂ ∉ free_evars ϕ
(∀ ρ : Valuation,
   eval ρ ϕ^{evar:0↦x₁} = ⊤ ∨ eval ρ ϕ^{evar:0↦x₁} = ∅)
→ ∀ ρ : Valuation,
    eval ρ ϕ^{evar:0↦x₂} = ⊤
    ∨ eval ρ ϕ^{evar:0↦x₂} = ∅
    intros H ρ.
signature: Signature
M: Model
x₁, x₂: evar
ϕ: Pattern
Hfr1: x₁ ∉ free_evars ϕ
Hfr2: x₂ ∉ free_evars ϕ
H: ∀ ρ : Valuation,
  eval ρ ϕ^{evar:0↦x₁} = ⊤
  ∨ eval ρ ϕ^{evar:0↦x₁} = ∅
ρ: Valuation
eval ρ ϕ^{evar:0↦x₂} = ⊤ ∨ eval ρ ϕ^{evar:0↦x₂} = ∅
    rewrite -(update_evar_val_same_2 x₂ ρ).
signature: Signature
M: Model
x₁, x₂: evar
ϕ: Pattern
Hfr1: x₁ ∉ free_evars ϕ
Hfr2: x₂ ∉ free_evars ϕ
H: ∀ ρ : Valuation,
  eval ρ ϕ^{evar:0↦x₁} = ⊤
  ∨ eval ρ ϕ^{evar:0↦x₁} = ∅
ρ: Valuation
eval (update_evar_val x₂ (evar_valuation ρ x₂) ρ)
  ϕ^{evar:0↦x₂} = ⊤
∨ eval (update_evar_val x₂ (evar_valuation ρ x₂) ρ)
    ϕ^{evar:0↦x₂} = ∅
    rewrite (eval_fresh_evar_open _ x₂ x₁); auto.
  Qed.

  Lemma M_predicate_evar_open_fresh_evar_2 x ϕ :
    evar_is_fresh_in x ϕ ->
    M_predicate M (ϕ^{evar: 0 ↦ (fresh_evar ϕ)}) ->
    M_predicate M (ϕ^{evar: 0 ↦ x}).
signature: Signature
M: Model
x: evar
ϕ: Pattern
evar_is_fresh_in x ϕ
→ M_predicate M ϕ^{evar:0↦fresh_evar ϕ}
  → M_predicate M ϕ^{evar:0↦x}
  Proof.
signature: Signature
M: Model
x: evar
ϕ: Pattern
evar_is_fresh_in x ϕ
→ M_predicate M ϕ^{evar:0↦fresh_evar ϕ}
  → M_predicate M ϕ^{evar:0↦x}
    intros Hfr H.
signature: Signature
M: Model
x: evar
ϕ: Pattern
Hfr: evar_is_fresh_in x ϕ
H: M_predicate M ϕ^{evar:0↦fresh_evar ϕ}
M_predicate M ϕ^{evar:0↦x}
    apply M_predicate_evar_open_fresh_evar_1 with (x₁ := fresh_evar ϕ); auto.
  Qed.

  (* Similar to plugging_patterns: using free svar substitution instead of 
     bound svar substitution.
     TODO: we may be able to gneeralize this lemma to non-closed psi,
           if we deal with nest_mu properly
   *)
  Lemma Private_free_svar_subst_update_exchange :
    ∀ sz phi psi X ρ,
      le (Pattern.size phi) sz → well_formed psi → well_formed_closed phi → 
      eval ρ (phi^[[svar: X ↦ psi]]) =
      eval (@update_svar_val M X (eval ρ psi) ρ) phi.
signature: Signature
M: Model
∀ (sz : nat) (phi psi : Pattern) (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ) phi
  Proof.
signature: Signature
M: Model
∀ (sz : nat) (phi psi : Pattern) (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ) phi
    unfold free_svar_subst.
signature: Signature
M: Model
∀ (sz : nat) (phi psi : Pattern) (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) (X0 : svar)
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex , free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu , free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ) phi
    induction sz; destruct phi; intros psi X ρ Hsz Hwf Hwfc ; simpl in *; try inversion Hsz; auto.
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
eval (update_svar_val X (eval ρ psi) ρ)
  (patt_free_svar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
eval (update_svar_val X (eval ρ psi) ρ)
  (patt_free_svar x)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 $
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ) (phi1 $ phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 $
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ) (phi1 $ phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 --->
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ)
  (phi1 ---> phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 --->
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ)
  (phi1 ---> phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
eval ρ
  (ex ,
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (ex , phi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
eval ρ
  (ex ,
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (ex , phi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
eval ρ
  (mu ,
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (mu , phi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
eval ρ
  (mu ,
   (fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (mu , phi)
    -
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
eval (update_svar_val X (eval ρ psi) ρ)
  (patt_free_svar x) rewrite -> eval_free_svar_simpl.
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
svar_valuation (update_svar_val X (eval ρ psi) ρ) x unfold update_svar_val.
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then eval ρ psi
        else svar_valuation ρ sv'
  |} x
      simpl.
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
(if decide (X = x)
 then eval ρ psi
 else svar_valuation ρ x)
      destruct (decide (X = x)); simpl.
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
e: X = x
eval ρ psi = eval ρ psi
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
n: X ≠ x
eval ρ (patt_free_svar x) = svar_valuation ρ x
      +
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
e: X = x
eval ρ psi = eval ρ psi reflexivity.
      +
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
n: X ≠ x
eval ρ (patt_free_svar x) = svar_valuation ρ x rewrite -> eval_free_svar_simpl.
signature: Signature
M: Model
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ 0
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
n: X ≠ x
svar_valuation ρ x = svar_valuation ρ x reflexivity.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
eval (update_svar_val X (eval ρ psi) ρ)
  (patt_free_svar x)  rewrite -> eval_free_svar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
svar_valuation (update_svar_val X (eval ρ psi) ρ) x unfold update_svar_val.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρ: Valuation
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
eval ρ
  (if decide (X = x) then psi else patt_free_svar x) =
svar_valuation
  {|
    evar_valuation := evar_valuation ρ;
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then eval ρ psi
        else svar_valuation ρ sv'
  |} x
      destruct ρ as [ρₑ ρₛ].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  (if decide (X = x) then psi else patt_free_svar x) =
svar_valuation
  {|
    evar_valuation :=
      evar_valuation
        {|
          evar_valuation := ρₑ; svar_valuation := ρₛ
        |};
    svar_valuation :=
      λ sv' : svar,
        if decide (X = sv')
        then
         eval
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} psi
        else
         svar_valuation
           {|
             evar_valuation := ρₑ;
             svar_valuation := ρₛ
           |} sv'
  |} x simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  (if decide (X = x) then psi else patt_free_svar x) =
(if decide (X = x)
 then
  eval
    {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
    psi
 else ρₛ x)
      destruct (decide (X = x)); simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
e: X = x
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  psi =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
n: X ≠ x
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  (patt_free_svar x) = 
ρₛ x
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
e: X = x
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  psi =
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  psi reflexivity.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
n: X ≠ x
eval {| evar_valuation := ρₑ; svar_valuation := ρₛ |}
  (patt_free_svar x) = ρₛ x rewrite -> eval_free_svar_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
x: svar
psi: Pattern
X: svar
ρₑ: evar → M
ρₛ: svar → propset M
Hsz: 0 ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (patt_free_svar x)
m: nat
H0: 0 ≤ sz
H: m = sz
n: X ≠ x
svar_valuation
  {| evar_valuation := ρₑ; svar_valuation := ρₛ |} x =
ρₛ x reflexivity.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 $
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ) (phi1 $ phi2) repeat rewrite -> eval_app_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
app_ext
  (eval ρ
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi1))
  (eval ρ
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi2)) =
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2)
      erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval ρ
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi2)) =
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2) =
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 $ phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 $ phi2) 0 =
true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed_mu_aux phi2 0 = true
∧ well_formed_closed_ex_aux phi2 0 = true split; auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz lia.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed psi auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 $ phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 $ phi2) 0 =
true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed_mu_aux phi1 0 = true
∧ well_formed_closed_ex_aux phi1 0 = true split; auto.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 $
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ) (phi1 $ phi2) repeat rewrite -> eval_app_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
app_ext
  (eval ρ
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi1))
  (eval ρ
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi2)) =
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2)
      erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval ρ
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi2)) =
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2) =
app_ext (eval (update_svar_val X (eval ρ psi) ρ) phi1)
  (eval (update_svar_val X (eval ρ psi) ρ) phi2)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
       +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 $ phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 $ phi2) 0 =
true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed_mu_aux phi2 0 = true
∧ well_formed_closed_ex_aux phi2 0 = true split; auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz lia.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 $ phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 $ phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 $ phi2) 0 =
true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed_mu_aux phi1 0 = true
∧ well_formed_closed_ex_aux phi1 0 = true split; auto.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 --->
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ)
  (phi1 ---> phi2) repeat rewrite -> eval_imp_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
⊤
∖ eval ρ
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi1)
∪ eval ρ
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi2) =
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2
      erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval ρ
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi2) =
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2 =
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
       +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi2 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 ---> phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 ---> phi2) 0 =
true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi2
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed_mu_aux phi2 0 = true
∧ well_formed_closed_ex_aux phi2 0 = true split; auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
size phi1 ≤ sz lia.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed psi auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 ---> phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 ---> phi2) 0 =
true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed phi1
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
H0: size phi1 + size phi2 = sz
well_formed_closed_mu_aux phi1 0 = true
∧ well_formed_closed_ex_aux phi1 0 = true split; auto.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
eval ρ
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi1 --->
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi2) =
eval (update_svar_val X (eval ρ psi) ρ)
  (phi1 ---> phi2) repeat rewrite -> eval_imp_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
⊤
∖ eval ρ
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi1)
∪ eval ρ
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi2) =
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2
      erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval ρ
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi2) =
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 erewrite -> IHsz.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2 =
⊤ ∖ eval (update_svar_val X (eval ρ psi) ρ) phi1
∪ eval (update_svar_val X (eval ρ psi) ρ) phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi2 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
       +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 ---> phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 ---> phi2) 0 =
true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi2
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed_mu_aux phi2 0 = true
∧ well_formed_closed_ex_aux phi2 0 = true split; auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
size phi1 ≤ sz lia.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed psi auto.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (phi1 ---> phi2)
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (phi1 ---> phi2) 0 =
true
Hwfc2: well_formed_closed_ex_aux (phi1 ---> phi2) 0 =
true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1 simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi1 0 &&
well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc1 as [Hwfc11 Hwfc12].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc2: well_formed_closed_ex_aux phi1 0 &&
well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
        apply andb_true_iff in Hwfc2 as [Hwfc21 Hwfc22].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed phi1
        apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi1, phi2, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi1 + size phi2) ≤ S sz
Hwf: well_formed psi
Hwfc11: well_formed_closed_mu_aux phi1 0 = true
Hwfc12: well_formed_closed_mu_aux phi2 0 = true
Hwfc21: well_formed_closed_ex_aux phi1 0 = true
Hwfc22: well_formed_closed_ex_aux phi2 0 = true
m: nat
H0: S (size phi1 + size phi2) ≤ sz
H: m = sz
well_formed_closed_mu_aux phi1 0 = true
∧ well_formed_closed_ex_aux phi1 0 = true split; auto.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
eval ρ
  (ex ,
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (ex , phi) repeat rewrite -> eval_ex_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
(let x :=
   fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi) in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ'
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x0 => patt_free_evar x0
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x0 => patt_bound_evar x0
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{evar:0↦x})) =
(let x := fresh_evar phi in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e
          (update_svar_val X (eval ρ psi) ρ) in
      eval ρ' phi^{evar:0↦x})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar
             ((fix free_svar_subst
                 (psi : Pattern) (X : svar)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x =>
                     patt_free_evar x
                 | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                 | patt_bound_evar x =>
                     patt_bound_evar x
                 | patt_bound_svar X' =>
                     patt_bound_svar X'
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                 | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                 | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                 end) psi X phi)) e ρ)
       ((fix free_svar_subst
           (psi : Pattern) (X : svar) (phi : Pattern)
           {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x => patt_free_evar x
           | patt_free_svar X' =>
               if decide (X = X')
               then psi
               else patt_free_svar X'
           | patt_bound_evar x => patt_bound_evar x
           | patt_bound_svar X' => patt_bound_svar X'
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               free_svar_subst psi X phi1 $
               free_svar_subst psi X phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               free_svar_subst psi X phi1 --->
               free_svar_subst psi X phi2
           | ex , phi' =>
               ex , free_svar_subst psi X phi'
           | mu , phi' =>
               mu , free_svar_subst psi X phi'
           end) psi X phi)^{evar:0↦fresh_evar
                                     ((fix
                                         free_svar_subst
                                         (psi : Pattern)
                                         (X : svar)
                                         (phi : Pattern)
                                         {struct phi} :
                                           Pattern :=
                                         match
                                           phi
                                         with
                                         | patt_free_evar
                                           x =>
                                             patt_free_evar
                                               x
                                         | patt_free_svar
                                           X' =>
                                             if
                                              decide
                                                (X =
                                                 X')
                                             then psi
                                             else
                                              patt_free_svar
                                                X'
                                         | patt_bound_evar
                                           x =>
                                             patt_bound_evar
                                               x
                                         | patt_bound_svar
                                           X' =>
                                             patt_bound_svar
                                               X'
                                         | patt_sym
                                           sigma =>
                                             patt_sym
                                               sigma
                                         | phi1 $ phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 $
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ⊥ => ⊥
                                         | phi1 --->
                                           phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 --->
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ex , phi' =>
                                             ex ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         | mu , phi' =>
                                             mu ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         end) psi X
                                        phi)}) =
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val (fresh_evar phi) e
          (update_svar_val X (eval ρ psi) ρ))
       phi^{evar:0↦fresh_evar phi})
      apply propset_fa_union_same.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
∀ c : M,
  eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} =
  eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c: M
eval
  (update_evar_val
     (fresh_evar
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x => patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x => patt_bound_evar x
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)) c ρ)
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦fresh_evar
                                ((fix free_svar_subst
                                    (psi : Pattern)
                                    (X : svar)
                                    (phi : Pattern)
                                    {struct phi} :
                                      Pattern :=
                                    match phi with
                                    | patt_free_evar x =>
                                        patt_free_evar
                                          x
                                    | patt_free_svar
                                      X' =>
                                        if
                                         decide
                                           (X = X')
                                        then psi
                                        else
                                         patt_free_svar
                                           X'
                                    | patt_bound_evar
                                      x =>
                                        patt_bound_evar
                                          x
                                    | patt_bound_svar
                                      X' =>
                                        patt_bound_svar
                                          X'
                                    | patt_sym sigma =>
                                        patt_sym sigma
                                    | phi1 $ phi2 =>
                                        free_svar_subst
                                          psi X phi1 $
                                        free_svar_subst
                                          psi X phi2
                                    | ⊥ => ⊥
                                    | phi1 ---> phi2 =>
                                        free_svar_subst
                                          psi X phi1 --->
                                        free_svar_subst
                                          psi X phi2
                                    | ex , phi' =>
                                        ex ,
                                        free_svar_subst
                                          psi X phi'
                                    | mu , phi' =>
                                        mu ,
                                        free_svar_subst
                                          psi X phi'
                                    end) psi X phi)} =
eval
  (update_evar_val (fresh_evar phi) c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦fresh_evar phi}
      rewrite -> set_eq_subseteq.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c: M
eval
  (update_evar_val
     (fresh_evar
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x => patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x => patt_bound_evar x
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)) c ρ)
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦fresh_evar
                                ((fix free_svar_subst
                                    (psi : Pattern)
                                    (X : svar)
                                    (phi : Pattern)
                                    {struct phi} :
                                      Pattern :=
                                    match phi with
                                    | patt_free_evar x =>
                                        patt_free_evar
                                          x
                                    | patt_free_svar
                                      X' =>
                                        if
                                         decide
                                           (X = X')
                                        then psi
                                        else
                                         patt_free_svar
                                           X'
                                    | patt_bound_evar
                                      x =>
                                        patt_bound_evar
                                          x
                                    | patt_bound_svar
                                      X' =>
                                        patt_bound_svar
                                          X'
                                    | patt_sym sigma =>
                                        patt_sym sigma
                                    | phi1 $ phi2 =>
                                        free_svar_subst
                                          psi X phi1 $
                                        free_svar_subst
                                          psi X phi2
                                    | ⊥ => ⊥
                                    | phi1 ---> phi2 =>
                                        free_svar_subst
                                          psi X phi1 --->
                                        free_svar_subst
                                          psi X phi2
                                    | ex , phi' =>
                                        ex ,
                                        free_svar_subst
                                          psi X phi'
                                    | mu , phi' =>
                                        mu ,
                                        free_svar_subst
                                          psi X phi'
                                    end) psi X phi)}
⊆ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
⊆ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
      repeat rewrite -> elem_of_subseteq.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c: M
(∀ x : M,
   x
   ∈ eval
       (update_evar_val
          (fresh_evar
             ((fix free_svar_subst
                 (psi : Pattern) (X : svar)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x0 =>
                     patt_free_evar x0
                 | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                 | patt_bound_evar x0 =>
                     patt_bound_evar x0
                 | patt_bound_svar X' =>
                     patt_bound_svar X'
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                 | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                 | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                 end) psi X phi)) c ρ)
       ((fix free_svar_subst
           (psi : Pattern) (X : svar) (phi : Pattern)
           {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x0 => patt_free_evar x0
           | patt_free_svar X' =>
               if decide (X = X')
               then psi
               else patt_free_svar X'
           | patt_bound_evar x0 => patt_bound_evar x0
           | patt_bound_svar X' => patt_bound_svar X'
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               free_svar_subst psi X phi1 $
               free_svar_subst psi X phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               free_svar_subst psi X phi1 --->
               free_svar_subst psi X phi2
           | ex , phi' =>
               ex , free_svar_subst psi X phi'
           | mu , phi' =>
               mu , free_svar_subst psi X phi'
           end) psi X phi)^{evar:0↦fresh_evar
                                     ((fix
                                         free_svar_subst
                                         (psi : Pattern)
                                         (X : svar)
                                         (phi : Pattern)
                                         {struct phi} :
                                           Pattern :=
                                         match
                                           phi
                                         with
                                         | patt_free_evar
                                           x0 =>
                                             patt_free_evar
                                               x0
                                         | patt_free_svar
                                           X' =>
                                             if
                                              decide
                                                (X =
                                                 X')
                                             then psi
                                             else
                                              patt_free_svar
                                                X'
                                         | patt_bound_evar
                                           x0 =>
                                             patt_bound_evar
                                               x0
                                         | patt_bound_svar
                                           X' =>
                                             patt_bound_svar
                                               X'
                                         | patt_sym
                                           sigma =>
                                             patt_sym
                                               sigma
                                         | phi1 $ phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 $
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ⊥ => ⊥
                                         | phi1 --->
                                           phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 --->
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ex , phi' =>
                                             ex ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         | mu , phi' =>
                                             mu ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         end) psi X
                                        phi)}
   → x
     ∈ eval
         (update_evar_val (fresh_evar phi) c
            (update_svar_val X (eval ρ psi) ρ))
         phi^{evar:0↦fresh_evar phi})
∧ (∀ x : M,
     x
     ∈ eval
         (update_evar_val (fresh_evar phi) c
            (update_svar_val X (eval ρ psi) ρ))
         phi^{evar:0↦fresh_evar phi}
     → x
       ∈ eval
           (update_evar_val
              (fresh_evar
                 ((fix free_svar_subst
                     (psi : Pattern) (X : svar)
                     (phi : Pattern) {struct phi} :
                       Pattern :=
                     match phi with
                     | patt_free_evar x0 =>
                         patt_free_evar x0
                     | patt_free_svar X' =>
                         if decide (X = X')
                         then psi
                         else patt_free_svar X'
                     | patt_bound_evar x0 =>
                         patt_bound_evar x0
                     | patt_bound_svar X' =>
                         patt_bound_svar X'
                     | patt_sym sigma =>
                         patt_sym sigma
                     | phi1 $ phi2 =>
                         free_svar_subst psi X phi1 $
                         free_svar_subst psi X phi2
                     | ⊥ => ⊥
                     | phi1 ---> phi2 =>
                         free_svar_subst psi X phi1 --->
                         free_svar_subst psi X phi2
                     | ex , phi' =>
                         ex ,
                         free_svar_subst psi X phi'
                     | mu , phi' =>
                         mu ,
                         free_svar_subst psi X phi'
                     end) psi X phi)) c ρ)
           ((fix free_svar_subst
               (psi : Pattern) (X : svar)
               (phi : Pattern) {struct phi} :
                 Pattern :=
               match phi with
               | patt_free_evar x0 =>
                   patt_free_evar x0
               | patt_free_svar X' =>
                   if decide (X = X')
                   then psi
                   else patt_free_svar X'
               | patt_bound_evar x0 =>
                   patt_bound_evar x0
               | patt_bound_svar X' =>
                   patt_bound_svar X'
               | patt_sym sigma => patt_sym sigma
               | phi1 $ phi2 =>
                   free_svar_subst psi X phi1 $
                   free_svar_subst psi X phi2
               | ⊥ => ⊥
               | phi1 ---> phi2 =>
                   free_svar_subst psi X phi1 --->
                   free_svar_subst psi X phi2
               | ex , phi' =>
                   ex , free_svar_subst psi X phi'
               | mu , phi' =>
                   mu , free_svar_subst psi X phi'
               end) psi X phi)^{evar:0↦fresh_evar
                                         ((fix
                                             free_svar_subst
                                             (psi : Pattern)
                                             (X : svar)
                                             (phi : Pattern)
                                             {struct
                                              phi} :
                                               Pattern :=
                                             match
                                               phi
                                             with
                                             | patt_free_evar
                                               x0 =>
                                                 patt_free_evar
                                                 x0
                                             | patt_free_svar
                                               X' =>
                                                 if
                                                 decide
                                                 (X =
                                                 X')
                                                 then
                                                 psi
                                                 else
                                                 patt_free_svar
                                                 X'
                                             | patt_bound_evar
                                               x0 =>
                                                 patt_bound_evar
                                                 x0
                                             | patt_bound_svar
                                               X' =>
                                                 patt_bound_svar
                                                 X'
                                             | patt_sym
                                               sigma =>
                                                 patt_sym
                                                 sigma
                                             | phi1 $
                                               phi2 =>
                                                 free_svar_subst
                                                 psi X
                                                 phi1 $
                                                 free_svar_subst
                                                 psi X
                                                 phi2
                                             | ⊥ => ⊥
                                             | phi1 --->
                                               phi2 =>
                                                 free_svar_subst
                                                 psi X
                                                 phi1 --->
                                                 free_svar_subst
                                                 psi X
                                                 phi2
                                             | ex ,
                                               phi' =>
                                                 ex ,
                                                 free_svar_subst
                                                 psi X
                                                 phi'
                                             | mu ,
                                               phi' =>
                                                 mu ,
                                                 free_svar_subst
                                                 psi X
                                                 phi'
                                             end) psi
                                            X phi)})
      split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val
         (fresh_evar
            ((fix free_svar_subst
                (psi : Pattern) (X : svar)
                (phi : Pattern) {struct phi} :
                  Pattern :=
                match phi with
                | patt_free_evar x0 =>
                    patt_free_evar x0
                | patt_free_svar X' =>
                    if decide (X = X')
                    then psi
                    else patt_free_svar X'
                | patt_bound_evar x0 =>
                    patt_bound_evar x0
                | patt_bound_svar X' =>
                    patt_bound_svar X'
                | patt_sym sigma => patt_sym sigma
                | phi1 $ phi2 =>
                    free_svar_subst psi X phi1 $
                    free_svar_subst psi X phi2
                | ⊥ => ⊥
                | phi1 ---> phi2 =>
                    free_svar_subst psi X phi1 --->
                    free_svar_subst psi X phi2
                | ex , phi' =>
                    ex , free_svar_subst psi X phi'
                | mu , phi' =>
                    mu , free_svar_subst psi X phi'
                end) psi X phi)) c ρ)
      ((fix free_svar_subst
          (psi : Pattern) (X : svar) (phi : Pattern)
          {struct phi} : Pattern :=
          match phi with
          | patt_free_evar x0 => patt_free_evar x0
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x0 => patt_bound_evar x0
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)^{evar:0↦fresh_evar
                                    ((fix
                                        free_svar_subst
                                        (psi : Pattern)
                                        (X : svar)
                                        (phi : Pattern)
                                        {struct phi} :
                                          Pattern :=
                                        match phi with
                                        | patt_free_evar
                                          x0 =>
                                            patt_free_evar
                                              x0
                                        | patt_free_svar
                                          X' =>
                                            if
                                             decide
                                               (X = X')
                                            then psi
                                            else
                                             patt_free_svar
                                               X'
                                        | patt_bound_evar
                                          x0 =>
                                            patt_bound_evar
                                              x0
                                        | patt_bound_svar
                                          X' =>
                                            patt_bound_svar
                                              X'
                                        | patt_sym
                                          sigma =>
                                            patt_sym
                                              sigma
                                        | phi1 $ phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 $
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ⊥ => ⊥
                                        | phi1 --->
                                          phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 --->
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ex , phi' =>
                                            ex ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        | mu , phi' =>
                                            mu ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        end) psi X phi)}
  → x
    ∈ eval
        (update_evar_val (fresh_evar phi) c
           (update_svar_val X (eval ρ psi) ρ))
        phi^{evar:0↦fresh_evar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val 
         (fresh_evar phi) c
         (update_svar_val X (eval ρ psi) ρ))
      phi^{evar:0↦fresh_evar phi}
  → x
    ∈ eval
        (update_evar_val
           (fresh_evar
              ((fix free_svar_subst
                  (psi : Pattern) 
                  (X : svar) 
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x0 =>
                     patt_free_evar x0
                  | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                  | patt_bound_evar x0 =>
                     patt_bound_evar x0
                  | patt_bound_svar X' =>
                     patt_bound_svar X'
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                  | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                  | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                  end) psi X phi)) c ρ)
        ((fix free_svar_subst
            (psi : Pattern) 
            (X : svar) 
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x0 => patt_free_evar x0
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x0 => patt_bound_evar x0
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{evar:0↦
        fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x0 => patt_free_evar x0
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x0 =>
                  patt_bound_evar x0
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)}
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val
         (fresh_evar
            ((fix free_svar_subst
                (psi : Pattern) (X : svar)
                (phi : Pattern) {struct phi} :
                  Pattern :=
                match phi with
                | patt_free_evar x0 =>
                    patt_free_evar x0
                | patt_free_svar X' =>
                    if decide (X = X')
                    then psi
                    else patt_free_svar X'
                | patt_bound_evar x0 =>
                    patt_bound_evar x0
                | patt_bound_svar X' =>
                    patt_bound_svar X'
                | patt_sym sigma => patt_sym sigma
                | phi1 $ phi2 =>
                    free_svar_subst psi X phi1 $
                    free_svar_subst psi X phi2
                | ⊥ => ⊥
                | phi1 ---> phi2 =>
                    free_svar_subst psi X phi1 --->
                    free_svar_subst psi X phi2
                | ex , phi' =>
                    ex , free_svar_subst psi X phi'
                | mu , phi' =>
                    mu , free_svar_subst psi X phi'
                end) psi X phi)) c ρ)
      ((fix free_svar_subst
          (psi : Pattern) (X : svar) (phi : Pattern)
          {struct phi} : Pattern :=
          match phi with
          | patt_free_evar x0 => patt_free_evar x0
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x0 => patt_bound_evar x0
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)^{evar:0↦fresh_evar
                                    ((fix
                                        free_svar_subst
                                        (psi : Pattern)
                                        (X : svar)
                                        (phi : Pattern)
                                        {struct phi} :
                                          Pattern :=
                                        match phi with
                                        | patt_free_evar
                                          x0 =>
                                            patt_free_evar
                                              x0
                                        | patt_free_svar
                                          X' =>
                                            if
                                             decide
                                               (X = X')
                                            then psi
                                            else
                                             patt_free_svar
                                               X'
                                        | patt_bound_evar
                                          x0 =>
                                            patt_bound_evar
                                              x0
                                        | patt_bound_svar
                                          X' =>
                                            patt_bound_svar
                                              X'
                                        | patt_sym
                                          sigma =>
                                            patt_sym
                                              sigma
                                        | phi1 $ phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 $
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ⊥ => ⊥
                                        | phi1 --->
                                          phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 --->
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ex , phi' =>
                                            ex ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        | mu , phi' =>
                                            mu ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        end) psi X phi)}
  → x
    ∈ eval
        (update_evar_val (fresh_evar phi) c
           (update_svar_val X (eval ρ psi) ρ))
        phi^{evar:0↦fresh_evar phi} intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember (evar_fresh  (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(evar_is_fresh_in (fresh_evar (phi^[[svar: X ↦ psi]])) (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(fresh ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: fresh ∉ B
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
          subst.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi0) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
      ∪ free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi apply set_evar_fresh_is_fresh'.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: fresh ∉ B
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        subst fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh (elements B) ∉ B
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} apply not_elem_of_union in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi)
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} destruct H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        apply not_elem_of_union in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]])
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars phi
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} destruct H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]]
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars phi
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H2: evar_fresh (elements B)
∉ free_evars phi^[[svar:X↦psi]]
H4: evar_fresh (elements B) ∉ free_evars phi
H3: evar_fresh (elements B) ∉ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember (evar_fresh (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(evar_is_fresh_in fresh (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
fresh ∉ free_evars phi^[[svar:X↦psi]] assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        erewrite eval_fresh_evar_open in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val ?y c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    ?y}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in ?y
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)
        3: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in ?y
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) eassumption. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)
        2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) assumption. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}

        assert(evar_is_fresh_in fresh phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
evar_is_fresh_in fresh phi
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
fresh ∉ free_evars phi assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(evar_is_fresh_in (fresh_evar phi) phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}

        epose (eval_fresh_evar_open _ _ _ c 0 (update_svar_val X (eval ρ psi) ρ) _ _) as HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
HFresh:= eval_fresh_evar_open ?ϕ 
  ?x ?y c 0 (update_svar_val X (eval ρ psi) ρ)
  ?Goal3 ?Goal4: eval
  (update_evar_val ?x c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?x} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?y}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        rewrite -> HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
HFresh:= eval_fresh_evar_open phi 
  (fresh_evar phi) ?y c 0
  (update_svar_val X (eval ρ psi) ρ) 
  ?Goal3 ?Goal4: eval
  (update_evar_val (fresh_evar phi) c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦fresh_evar phi} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦?y}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        clear HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        fold free_svar_subst in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        epose proof (IHsz (phi^{evar: 0 ↦ fresh}) 
                    psi X
                    (update_evar_val fresh c ρ) _ _ ) as H8.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    phi^{evar:0↦fresh}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        feed specialize H8.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    phi^{evar:0↦fresh}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    phi^{evar:0↦fresh}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh} wf_auto2. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        pose proof (eval_free_evar_independent ρ fresh c psi) as H9.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        rewrite -> H9 in H8.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi clear H9.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        unfold free_svar_subst in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        rewrite -> (@evar_open_free_svar_subst_comm) in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^{evar:0↦fresh}^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        unfold free_svar_subst in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        rewrite -> H8 in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  phi^{evar:0↦fresh}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_evar_val fresh c ρ) psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
H9: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        all: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y} exact H.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' =>
             patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
B: EVarSet
HeqB: B =
free_evars
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi) ∪ 
free_evars phi ∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh
∉ free_evars
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H8: eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi^{evar:0↦fresh}) =
eval
  (update_svar_val X (eval ρ psi)
     (update_evar_val fresh c ρ))
  phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 wf_auto2.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val (fresh_evar phi) c
         (update_svar_val X (eval ρ psi) ρ))
      phi^{evar:0↦fresh_evar phi}
  → x
    ∈ eval
        (update_evar_val
           (fresh_evar
              ((fix free_svar_subst
                  (psi : Pattern) (X : svar)
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x0 =>
                      patt_free_evar x0
                  | patt_free_svar X' =>
                      if decide (X = X')
                      then psi
                      else patt_free_svar X'
                  | patt_bound_evar x0 =>
                      patt_bound_evar x0
                  | patt_bound_svar X' =>
                      patt_bound_svar X'
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                      free_svar_subst psi X phi1 $
                      free_svar_subst psi X phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                      free_svar_subst psi X phi1 --->
                      free_svar_subst psi X phi2
                  | ex , phi' =>
                      ex , free_svar_subst psi X phi'
                  | mu , phi' =>
                      mu , free_svar_subst psi X phi'
                  end) psi X phi)) c ρ)
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x0 => patt_free_evar x0
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x0 => patt_bound_evar x0
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{evar:0↦fresh_evar
                                      ((fix
                                          free_svar_subst
                                          (psi : Pattern)
                                          (X : svar)
                                          (phi : Pattern)
                                          {struct phi} :
                                            Pattern :=
                                          match
                                            phi
                                          with
                                          | patt_free_evar
                                            x0 =>
                                              patt_free_evar
                                                x0
                                          | patt_free_svar
                                            X' =>
                                              if
                                               decide
                                                 (X =
                                                 X')
                                              then psi
                                              else
                                               patt_free_svar
                                                 X'
                                          | patt_bound_evar
                                            x0 =>
                                              patt_bound_evar
                                                x0
                                          | patt_bound_svar
                                            X' =>
                                              patt_bound_svar
                                                X'
                                          | patt_sym
                                            sigma =>
                                              patt_sym
                                                sigma
                                          | phi1 $
                                            phi2 =>
                                              free_svar_subst
                                                psi X
                                                phi1 $
                                              free_svar_subst
                                                psi X
                                                phi2
                                          | ⊥ => ⊥
                                          | phi1 --->
                                            phi2 =>
                                              free_svar_subst
                                                psi X
                                                phi1 --->
                                              free_svar_subst
                                                psi X
                                                phi2
                                          | ex , phi' =>
                                              ex ,
                                              free_svar_subst
                                                psi X
                                                phi'
                                          | mu , phi' =>
                                              mu ,
                                              free_svar_subst
                                                psi X
                                                phi'
                                          end) psi X
                                         phi)} intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember (evar_fresh (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        assert(evar_is_fresh_in (fresh_evar (phi^[[svar: X ↦ psi]])) (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        assert(fresh ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: fresh ∉ B
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
          subst.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi0) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
      ∪ free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi apply set_evar_fresh_is_fresh'.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: fresh ∉ B
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        subst fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh (elements B) ∉ B
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} apply not_elem_of_union in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi)
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} destruct H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        apply not_elem_of_union in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]])
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars phi
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} destruct H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H2: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]]
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars phi
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H2: evar_fresh (elements B)
∉ free_evars phi^[[svar:X↦psi]]
H4: evar_fresh (elements B) ∉ free_evars phi
H3: evar_fresh (elements B) ∉ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember (evar_fresh (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        assert(evar_is_fresh_in fresh (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
fresh ∉ free_evars phi^[[svar:X↦psi]] assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        erewrite -> eval_fresh_evar_open.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval (update_evar_val ?y c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in ?y
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)
        3: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in ?y
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) eassumption. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) 
         (X : svar) (phi : Pattern) {struct phi} :
           Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)
        2: {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in
  (fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi))
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) assumption. }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        assert(evar_is_fresh_in fresh phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in fresh phi
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
fresh ∉ free_evars phi assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        assert(evar_is_fresh_in (fresh_evar phi) phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        epose (eval_fresh_evar_open _ _ _ c 0 (update_svar_val X (eval ρ psi) ρ) _ _) as HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
HFresh:= eval_fresh_evar_open ?ϕ 
  ?x ?y c 0 (update_svar_val X (eval ρ psi) ρ)
  ?Goal6 ?Goal7: eval
  (update_evar_val ?x c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?x} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?y}
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        rewrite -> HFresh in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
HFresh:= eval_fresh_evar_open phi 
  (fresh_evar phi) ?y c 0
  (update_svar_val X (eval ρ psi) ρ) 
  ?Goal6 ?Goal7: eval
  (update_evar_val (fresh_evar phi) c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦fresh_evar phi} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦?y}
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        clear HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        epose (IHsz (phi^{evar: 0 ↦ fresh}) 
                    psi X
                    (update_evar_val fresh c ρ) _ _ ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        pose (eval_free_evar_independent ρ fresh c psi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
        rewrite -> e0 in e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi clear e0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi fold free_svar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi

        rewrite -> (evar_open_free_svar_subst_comm).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    phi^{evar:0↦fresh}^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        unfold free_svar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        rewrite -> e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_svar_val X (eval ρ psi)
       (update_evar_val fresh c ρ)) phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal8 ?Goal9: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        all: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_svar_val X (eval ρ psi)
       (update_evar_val fresh c ρ)) phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_svar_val X (eval ρ psi)
       (update_evar_val fresh c ρ)) phi^{evar:0↦fresh} exact H.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh} apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (ex , phi) 0 = true
Hwfc2: well_formed_closed_ex_aux (ex , phi) 0 = true
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh} simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
          apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true
∧ well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 =
  true split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 = true
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true now apply wfc_mu_aux_body_ex_imp1.
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 = true now apply wfc_ex_aux_body_ex_imp1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply andb_true_iff in Hwf as [_ Hwf].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed psi = true
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply andb_true_iff in Hwf as [_ Hwf].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed_ex_aux psi 0 = true
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply Hwf.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
eval ρ
  (ex ,
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (ex , phi) repeat rewrite -> eval_ex_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
(let x :=
   fresh_evar
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi) in
 propset_fa_union
   (λ e : M,
      let ρ' := update_evar_val x e ρ in
      eval ρ'
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x0 => patt_free_evar x0
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x0 => patt_bound_evar x0
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{evar:0↦x})) =
(let x := fresh_evar phi in
 propset_fa_union
   (λ e : M,
      let ρ' :=
        update_evar_val x e
          (update_svar_val X (eval ρ psi) ρ) in
      eval ρ' phi^{evar:0↦x})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val
          (fresh_evar
             ((fix free_svar_subst
                 (psi : Pattern) (X : svar)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x =>
                     patt_free_evar x
                 | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                 | patt_bound_evar x =>
                     patt_bound_evar x
                 | patt_bound_svar X' =>
                     patt_bound_svar X'
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                 | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                 | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                 end) psi X phi)) e ρ)
       ((fix free_svar_subst
           (psi : Pattern) (X : svar) (phi : Pattern)
           {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x => patt_free_evar x
           | patt_free_svar X' =>
               if decide (X = X')
               then psi
               else patt_free_svar X'
           | patt_bound_evar x => patt_bound_evar x
           | patt_bound_svar X' => patt_bound_svar X'
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               free_svar_subst psi X phi1 $
               free_svar_subst psi X phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               free_svar_subst psi X phi1 --->
               free_svar_subst psi X phi2
           | ex , phi' =>
               ex , free_svar_subst psi X phi'
           | mu , phi' =>
               mu , free_svar_subst psi X phi'
           end) psi X phi)^{evar:0↦fresh_evar
                                     ((fix
                                         free_svar_subst
                                         (psi : Pattern)
                                         (X : svar)
                                         (phi : Pattern)
                                         {struct phi} :
                                           Pattern :=
                                         match
                                           phi
                                         with
                                         | patt_free_evar
                                           x =>
                                             patt_free_evar
                                               x
                                         | patt_free_svar
                                           X' =>
                                             if
                                              decide
                                                (X =
                                                 X')
                                             then psi
                                             else
                                              patt_free_svar
                                                X'
                                         | patt_bound_evar
                                           x =>
                                             patt_bound_evar
                                               x
                                         | patt_bound_svar
                                           X' =>
                                             patt_bound_svar
                                               X'
                                         | patt_sym
                                           sigma =>
                                             patt_sym
                                               sigma
                                         | phi1 $ phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 $
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ⊥ => ⊥
                                         | phi1 --->
                                           phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 --->
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ex , phi' =>
                                             ex ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         | mu , phi' =>
                                             mu ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         end) psi X
                                        phi)}) =
propset_fa_union
  (λ e : M,
     eval
       (update_evar_val (fresh_evar phi) e
          (update_svar_val X (eval ρ psi) ρ))
       phi^{evar:0↦fresh_evar phi})
      apply propset_fa_union_same.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
∀ c : M,
  eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} =
  eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c: M
eval
  (update_evar_val
     (fresh_evar
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x => patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x => patt_bound_evar x
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)) c ρ)
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦fresh_evar
                                ((fix free_svar_subst
                                    (psi : Pattern)
                                    (X : svar)
                                    (phi : Pattern)
                                    {struct phi} :
                                      Pattern :=
                                    match phi with
                                    | patt_free_evar x =>
                                        patt_free_evar
                                          x
                                    | patt_free_svar
                                      X' =>
                                        if
                                         decide
                                           (X = X')
                                        then psi
                                        else
                                         patt_free_svar
                                           X'
                                    | patt_bound_evar
                                      x =>
                                        patt_bound_evar
                                          x
                                    | patt_bound_svar
                                      X' =>
                                        patt_bound_svar
                                          X'
                                    | patt_sym sigma =>
                                        patt_sym sigma
                                    | phi1 $ phi2 =>
                                        free_svar_subst
                                          psi X phi1 $
                                        free_svar_subst
                                          psi X phi2
                                    | ⊥ => ⊥
                                    | phi1 ---> phi2 =>
                                        free_svar_subst
                                          psi X phi1 --->
                                        free_svar_subst
                                          psi X phi2
                                    | ex , phi' =>
                                        ex ,
                                        free_svar_subst
                                          psi X phi'
                                    | mu , phi' =>
                                        mu ,
                                        free_svar_subst
                                          psi X phi'
                                    end) psi X phi)} =
eval
  (update_evar_val (fresh_evar phi) c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦fresh_evar phi}
      rewrite -> set_eq_subseteq.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c: M
eval
  (update_evar_val
     (fresh_evar
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x => patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x => patt_bound_evar x
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)) c ρ)
  ((fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦fresh_evar
                                ((fix free_svar_subst
                                    (psi : Pattern)
                                    (X : svar)
                                    (phi : Pattern)
                                    {struct phi} :
                                      Pattern :=
                                    match phi with
                                    | patt_free_evar x =>
                                        patt_free_evar
                                          x
                                    | patt_free_svar
                                      X' =>
                                        if
                                         decide
                                           (X = X')
                                        then psi
                                        else
                                         patt_free_svar
                                           X'
                                    | patt_bound_evar
                                      x =>
                                        patt_bound_evar
                                          x
                                    | patt_bound_svar
                                      X' =>
                                        patt_bound_svar
                                          X'
                                    | patt_sym sigma =>
                                        patt_sym sigma
                                    | phi1 $ phi2 =>
                                        free_svar_subst
                                          psi X phi1 $
                                        free_svar_subst
                                          psi X phi2
                                    | ⊥ => ⊥
                                    | phi1 ---> phi2 =>
                                        free_svar_subst
                                          psi X phi1 --->
                                        free_svar_subst
                                          psi X phi2
                                    | ex , phi' =>
                                        ex ,
                                        free_svar_subst
                                          psi X phi'
                                    | mu , phi' =>
                                        mu ,
                                        free_svar_subst
                                          psi X phi'
                                    end) psi X phi)}
⊆ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
⊆ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
      repeat rewrite -> elem_of_subseteq.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c: M
(∀ x : M,
   x
   ∈ eval
       (update_evar_val
          (fresh_evar
             ((fix free_svar_subst
                 (psi : Pattern) (X : svar)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x0 =>
                     patt_free_evar x0
                 | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                 | patt_bound_evar x0 =>
                     patt_bound_evar x0
                 | patt_bound_svar X' =>
                     patt_bound_svar X'
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                 | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                 | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                 end) psi X phi)) c ρ)
       ((fix free_svar_subst
           (psi : Pattern) (X : svar) (phi : Pattern)
           {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x0 => patt_free_evar x0
           | patt_free_svar X' =>
               if decide (X = X')
               then psi
               else patt_free_svar X'
           | patt_bound_evar x0 => patt_bound_evar x0
           | patt_bound_svar X' => patt_bound_svar X'
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               free_svar_subst psi X phi1 $
               free_svar_subst psi X phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               free_svar_subst psi X phi1 --->
               free_svar_subst psi X phi2
           | ex , phi' =>
               ex , free_svar_subst psi X phi'
           | mu , phi' =>
               mu , free_svar_subst psi X phi'
           end) psi X phi)^{evar:0↦fresh_evar
                                     ((fix
                                         free_svar_subst
                                         (psi : Pattern)
                                         (X : svar)
                                         (phi : Pattern)
                                         {struct phi} :
                                           Pattern :=
                                         match
                                           phi
                                         with
                                         | patt_free_evar
                                           x0 =>
                                             patt_free_evar
                                               x0
                                         | patt_free_svar
                                           X' =>
                                             if
                                              decide
                                                (X =
                                                 X')
                                             then psi
                                             else
                                              patt_free_svar
                                                X'
                                         | patt_bound_evar
                                           x0 =>
                                             patt_bound_evar
                                               x0
                                         | patt_bound_svar
                                           X' =>
                                             patt_bound_svar
                                               X'
                                         | patt_sym
                                           sigma =>
                                             patt_sym
                                               sigma
                                         | phi1 $ phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 $
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ⊥ => ⊥
                                         | phi1 --->
                                           phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 --->
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ex , phi' =>
                                             ex ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         | mu , phi' =>
                                             mu ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         end) psi X
                                        phi)}
   → x
     ∈ eval
         (update_evar_val (fresh_evar phi) c
            (update_svar_val X (eval ρ psi) ρ))
         phi^{evar:0↦fresh_evar phi})
∧ (∀ x : M,
     x
     ∈ eval
         (update_evar_val (fresh_evar phi) c
            (update_svar_val X (eval ρ psi) ρ))
         phi^{evar:0↦fresh_evar phi}
     → x
       ∈ eval
           (update_evar_val
              (fresh_evar
                 ((fix free_svar_subst
                     (psi : Pattern) (X : svar)
                     (phi : Pattern) {struct phi} :
                       Pattern :=
                     match phi with
                     | patt_free_evar x0 =>
                         patt_free_evar x0
                     | patt_free_svar X' =>
                         if decide (X = X')
                         then psi
                         else patt_free_svar X'
                     | patt_bound_evar x0 =>
                         patt_bound_evar x0
                     | patt_bound_svar X' =>
                         patt_bound_svar X'
                     | patt_sym sigma =>
                         patt_sym sigma
                     | phi1 $ phi2 =>
                         free_svar_subst psi X phi1 $
                         free_svar_subst psi X phi2
                     | ⊥ => ⊥
                     | phi1 ---> phi2 =>
                         free_svar_subst psi X phi1 --->
                         free_svar_subst psi X phi2
                     | ex , phi' =>
                         ex ,
                         free_svar_subst psi X phi'
                     | mu , phi' =>
                         mu ,
                         free_svar_subst psi X phi'
                     end) psi X phi)) c ρ)
           ((fix free_svar_subst
               (psi : Pattern) (X : svar)
               (phi : Pattern) {struct phi} :
                 Pattern :=
               match phi with
               | patt_free_evar x0 =>
                   patt_free_evar x0
               | patt_free_svar X' =>
                   if decide (X = X')
                   then psi
                   else patt_free_svar X'
               | patt_bound_evar x0 =>
                   patt_bound_evar x0
               | patt_bound_svar X' =>
                   patt_bound_svar X'
               | patt_sym sigma => patt_sym sigma
               | phi1 $ phi2 =>
                   free_svar_subst psi X phi1 $
                   free_svar_subst psi X phi2
               | ⊥ => ⊥
               | phi1 ---> phi2 =>
                   free_svar_subst psi X phi1 --->
                   free_svar_subst psi X phi2
               | ex , phi' =>
                   ex , free_svar_subst psi X phi'
               | mu , phi' =>
                   mu , free_svar_subst psi X phi'
               end) psi X phi)^{evar:0↦fresh_evar
                                         ((fix
                                             free_svar_subst
                                             (psi : Pattern)
                                             (X : svar)
                                             (phi : Pattern)
                                             {struct
                                              phi} :
                                               Pattern :=
                                             match
                                               phi
                                             with
                                             | patt_free_evar
                                               x0 =>
                                                 patt_free_evar
                                                 x0
                                             | patt_free_svar
                                               X' =>
                                                 if
                                                 decide
                                                 (X =
                                                 X')
                                                 then
                                                 psi
                                                 else
                                                 patt_free_svar
                                                 X'
                                             | patt_bound_evar
                                               x0 =>
                                                 patt_bound_evar
                                                 x0
                                             | patt_bound_svar
                                               X' =>
                                                 patt_bound_svar
                                                 X'
                                             | patt_sym
                                               sigma =>
                                                 patt_sym
                                                 sigma
                                             | phi1 $
                                               phi2 =>
                                                 free_svar_subst
                                                 psi X
                                                 phi1 $
                                                 free_svar_subst
                                                 psi X
                                                 phi2
                                             | ⊥ => ⊥
                                             | phi1 --->
                                               phi2 =>
                                                 free_svar_subst
                                                 psi X
                                                 phi1 --->
                                                 free_svar_subst
                                                 psi X
                                                 phi2
                                             | ex ,
                                               phi' =>
                                                 ex ,
                                                 free_svar_subst
                                                 psi X
                                                 phi'
                                             | mu ,
                                               phi' =>
                                                 mu ,
                                                 free_svar_subst
                                                 psi X
                                                 phi'
                                             end) psi
                                            X phi)})
      split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val
         (fresh_evar
            ((fix free_svar_subst
                (psi : Pattern) (X : svar)
                (phi : Pattern) {struct phi} :
                  Pattern :=
                match phi with
                | patt_free_evar x0 =>
                    patt_free_evar x0
                | patt_free_svar X' =>
                    if decide (X = X')
                    then psi
                    else patt_free_svar X'
                | patt_bound_evar x0 =>
                    patt_bound_evar x0
                | patt_bound_svar X' =>
                    patt_bound_svar X'
                | patt_sym sigma => patt_sym sigma
                | phi1 $ phi2 =>
                    free_svar_subst psi X phi1 $
                    free_svar_subst psi X phi2
                | ⊥ => ⊥
                | phi1 ---> phi2 =>
                    free_svar_subst psi X phi1 --->
                    free_svar_subst psi X phi2
                | ex , phi' =>
                    ex , free_svar_subst psi X phi'
                | mu , phi' =>
                    mu , free_svar_subst psi X phi'
                end) psi X phi)) c ρ)
      ((fix free_svar_subst
          (psi : Pattern) (X : svar) (phi : Pattern)
          {struct phi} : Pattern :=
          match phi with
          | patt_free_evar x0 => patt_free_evar x0
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x0 => patt_bound_evar x0
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)^{evar:0↦fresh_evar
                                    ((fix
                                        free_svar_subst
                                        (psi : Pattern)
                                        (X : svar)
                                        (phi : Pattern)
                                        {struct phi} :
                                          Pattern :=
                                        match phi with
                                        | patt_free_evar
                                          x0 =>
                                            patt_free_evar
                                              x0
                                        | patt_free_svar
                                          X' =>
                                            if
                                             decide
                                               (X = X')
                                            then psi
                                            else
                                             patt_free_svar
                                               X'
                                        | patt_bound_evar
                                          x0 =>
                                            patt_bound_evar
                                              x0
                                        | patt_bound_svar
                                          X' =>
                                            patt_bound_svar
                                              X'
                                        | patt_sym
                                          sigma =>
                                            patt_sym
                                              sigma
                                        | phi1 $ phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 $
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ⊥ => ⊥
                                        | phi1 --->
                                          phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 --->
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ex , phi' =>
                                            ex ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        | mu , phi' =>
                                            mu ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        end) psi X phi)}
  → x
    ∈ eval
        (update_evar_val (fresh_evar phi) c
           (update_svar_val X (eval ρ psi) ρ))
        phi^{evar:0↦fresh_evar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val 
         (fresh_evar phi) c
         (update_svar_val X (eval ρ psi) ρ))
      phi^{evar:0↦fresh_evar phi}
  → x
    ∈ eval
        (update_evar_val
           (fresh_evar
              ((fix free_svar_subst
                  (psi : Pattern) 
                  (X : svar) 
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x0 =>
                     patt_free_evar x0
                  | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                  | patt_bound_evar x0 =>
                     patt_bound_evar x0
                  | patt_bound_svar X' =>
                     patt_bound_svar X'
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                  | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                  | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                  end) psi X phi)) c ρ)
        ((fix free_svar_subst
            (psi : Pattern) 
            (X : svar) 
            (phi : Pattern) {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x0 => patt_free_evar x0
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x0 => patt_bound_evar x0
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{evar:0↦
        fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x0 => patt_free_evar x0
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x0 =>
                  patt_bound_evar x0
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)}
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val
         (fresh_evar
            ((fix free_svar_subst
                (psi : Pattern) (X : svar)
                (phi : Pattern) {struct phi} :
                  Pattern :=
                match phi with
                | patt_free_evar x0 =>
                    patt_free_evar x0
                | patt_free_svar X' =>
                    if decide (X = X')
                    then psi
                    else patt_free_svar X'
                | patt_bound_evar x0 =>
                    patt_bound_evar x0
                | patt_bound_svar X' =>
                    patt_bound_svar X'
                | patt_sym sigma => patt_sym sigma
                | phi1 $ phi2 =>
                    free_svar_subst psi X phi1 $
                    free_svar_subst psi X phi2
                | ⊥ => ⊥
                | phi1 ---> phi2 =>
                    free_svar_subst psi X phi1 --->
                    free_svar_subst psi X phi2
                | ex , phi' =>
                    ex , free_svar_subst psi X phi'
                | mu , phi' =>
                    mu , free_svar_subst psi X phi'
                end) psi X phi)) c ρ)
      ((fix free_svar_subst
          (psi : Pattern) (X : svar) (phi : Pattern)
          {struct phi} : Pattern :=
          match phi with
          | patt_free_evar x0 => patt_free_evar x0
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x0 => patt_bound_evar x0
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)^{evar:0↦fresh_evar
                                    ((fix
                                        free_svar_subst
                                        (psi : Pattern)
                                        (X : svar)
                                        (phi : Pattern)
                                        {struct phi} :
                                          Pattern :=
                                        match phi with
                                        | patt_free_evar
                                          x0 =>
                                            patt_free_evar
                                              x0
                                        | patt_free_svar
                                          X' =>
                                            if
                                             decide
                                               (X = X')
                                            then psi
                                            else
                                             patt_free_svar
                                               X'
                                        | patt_bound_evar
                                          x0 =>
                                            patt_bound_evar
                                              x0
                                        | patt_bound_svar
                                          X' =>
                                            patt_bound_svar
                                              X'
                                        | patt_sym
                                          sigma =>
                                            patt_sym
                                              sigma
                                        | phi1 $ phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 $
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ⊥ => ⊥
                                        | phi1 --->
                                          phi2 =>
                                            free_svar_subst
                                              psi X
                                              phi1 --->
                                            free_svar_subst
                                              psi X
                                              phi2
                                        | ex , phi' =>
                                            ex ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        | mu , phi' =>
                                            mu ,
                                            free_svar_subst
                                              psi X
                                              phi'
                                        end) psi X phi)}
  → x
    ∈ eval
        (update_evar_val (fresh_evar phi) c
           (update_svar_val X (eval ρ psi) ρ))
        phi^{evar:0↦fresh_evar phi} intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember (evar_fresh (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(evar_is_fresh_in (fresh_evar (phi^[[svar: X ↦ psi]])) (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(fresh ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: fresh ∉ B
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
          subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: S (size phi) ≤ sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
      ∪ free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi apply set_evar_fresh_is_fresh'.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: fresh ∉ B
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        subst fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh (elements B) ∉ B
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} apply not_elem_of_union in H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi)
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} destruct H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        apply not_elem_of_union in H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]])
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars phi
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi} destruct H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]]
H5: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars phi
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H3: evar_fresh (elements B)
∉ free_evars phi^[[svar:X↦psi]]
H5: evar_fresh (elements B) ∉ free_evars phi
H4: evar_fresh (elements B) ∉ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        remember (evar_fresh (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(evar_is_fresh_in fresh (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
fresh ∉ free_evars phi^[[svar:X↦psi]] assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}

        epose proof (eval_fresh_evar_open _ _ _ c 0 ρ
                                                    H2 H3) as HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
HFresh: eval
  (update_evar_val
     (fresh_evar phi^[[svar:X↦psi]]) c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦
  fresh_evar phi^[[svar:X↦psi]]} =
eval (update_evar_val fresh c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦fresh}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        unfold free_svar_subst in HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x =>
              patt_bound_evar x
          | patt_bound_svar X' =>
              patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
HFresh: eval
  (update_evar_val
     (fresh_evar
        ((fix free_svar_subst
            (psi : Pattern) 
            (X : svar) 
            (phi : Pattern) {struct phi} :
              Pattern :=
            match phi with
            | patt_free_evar x =>
                patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x =>
                patt_bound_evar x
            | patt_bound_svar X' =>
                patt_bound_svar X'
            | patt_sym sigma =>
                patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex ,
                free_svar_subst psi X phi'
            | mu , phi' =>
                mu ,
                free_svar_subst psi X phi'
            end) psi X phi)) c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x =>
          patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦
  fresh_evar
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct
         phi} : Pattern :=
        match phi with
        | patt_free_evar x =>
            patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x =>
            patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)} =
eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x =>
          patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦fresh}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        rewrite -> HFresh in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
HFresh: eval
  (update_evar_val
     (fresh_evar
        ((fix free_svar_subst
            (psi : Pattern) 
            (X : svar) 
            (phi : Pattern) {struct phi} :
              Pattern :=
            match phi with
            | patt_free_evar x =>
                patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x =>
                patt_bound_evar x
            | patt_bound_svar X' =>
                patt_bound_svar X'
            | patt_sym sigma =>
                patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex ,
                free_svar_subst psi X phi'
            | mu , phi' =>
                mu ,
                free_svar_subst psi X phi'
            end) psi X phi)) c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x =>
          patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦
  fresh_evar
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct
         phi} : Pattern :=
        match phi with
        | patt_free_evar x =>
            patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x =>
            patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)} =
eval (update_evar_val fresh c ρ)
  ((fix free_svar_subst
      (psi : Pattern) 
      (X : svar) (phi : Pattern) {struct phi} :
        Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x =>
          patt_bound_evar x
      | patt_bound_svar X' =>
          patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' =>
          ex , free_svar_subst psi X phi'
      | mu , phi' =>
          mu , free_svar_subst psi X phi'
      end) psi X phi)^{evar:0↦fresh}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        clear HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(evar_is_fresh_in fresh phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in fresh phi
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
fresh ∉ free_evars phi assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        assert(evar_is_fresh_in (fresh_evar phi) phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        epose (eval_fresh_evar_open _ _ _ c 0 (update_svar_val X (eval ρ psi) ρ) _ _) as HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
HFresh:= eval_fresh_evar_open ?ϕ 
  ?x ?y c 0 (update_svar_val X (eval ρ psi) ρ)
  ?Goal10 ?Goal11: eval
  (update_evar_val ?x c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?x} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?y}
x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
        rewrite -> HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
HFresh:= eval_fresh_evar_open phi 
  (fresh_evar phi) ?y c 0
  (update_svar_val X (eval ρ psi) ρ) 
  ?Goal10 ?Goal11: eval
  (update_evar_val (fresh_evar phi) c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦fresh_evar phi} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦?y}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        clear HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        epose (IHsz (phi^{evar: 0 ↦ fresh})
                    psi X
                    (update_evar_val fresh c ρ) _ _ ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        pose (eval_free_evar_independent ρ fresh c psi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
        rewrite -> e0 in e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi clear e0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi fold free_svar_subst in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi

        rewrite -> (evar_open_free_svar_subst_comm) in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^{evar:0↦fresh}^[[svar:X↦psi]]
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        unfold free_svar_subst in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        rewrite -> e in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal12 ?Goal13: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        all: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y} exact H1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh} apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (ex , phi) 0 = true
Hwfc2: well_formed_closed_ex_aux (ex , phi) 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh} simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
          apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true
∧ well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 =
  true split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 = true
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true now apply wfc_mu_aux_body_ex_imp1.
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 = true now apply wfc_ex_aux_body_ex_imp1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply andb_true_iff in Hwf as [_ Hwf].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed psi = true
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply andb_true_iff in Hwf as [_ Hwf].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed_ex_aux psi 0 = true
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply Hwf.
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c: M
∀ x : M,
  x
  ∈ eval
      (update_evar_val (fresh_evar phi) c
         (update_svar_val X (eval ρ psi) ρ))
      phi^{evar:0↦fresh_evar phi}
  → x
    ∈ eval
        (update_evar_val
           (fresh_evar
              ((fix free_svar_subst
                  (psi : Pattern) (X : svar)
                  (phi : Pattern) {struct phi} :
                    Pattern :=
                  match phi with
                  | patt_free_evar x0 =>
                      patt_free_evar x0
                  | patt_free_svar X' =>
                      if decide (X = X')
                      then psi
                      else patt_free_svar X'
                  | patt_bound_evar x0 =>
                      patt_bound_evar x0
                  | patt_bound_svar X' =>
                      patt_bound_svar X'
                  | patt_sym sigma => patt_sym sigma
                  | phi1 $ phi2 =>
                      free_svar_subst psi X phi1 $
                      free_svar_subst psi X phi2
                  | ⊥ => ⊥
                  | phi1 ---> phi2 =>
                      free_svar_subst psi X phi1 --->
                      free_svar_subst psi X phi2
                  | ex , phi' =>
                      ex , free_svar_subst psi X phi'
                  | mu , phi' =>
                      mu , free_svar_subst psi X phi'
                  end) psi X phi)) c ρ)
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x0 => patt_free_evar x0
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x0 => patt_bound_evar x0
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{evar:0↦fresh_evar
                                      ((fix
                                          free_svar_subst
                                          (psi : Pattern)
                                          (X : svar)
                                          (phi : Pattern)
                                          {struct phi} :
                                            Pattern :=
                                          match
                                            phi
                                          with
                                          | patt_free_evar
                                            x0 =>
                                              patt_free_evar
                                                x0
                                          | patt_free_svar
                                            X' =>
                                              if
                                               decide
                                                 (X =
                                                 X')
                                              then psi
                                              else
                                               patt_free_svar
                                                 X'
                                          | patt_bound_evar
                                            x0 =>
                                              patt_bound_evar
                                                x0
                                          | patt_bound_svar
                                            X' =>
                                              patt_bound_svar
                                                X'
                                          | patt_sym
                                            sigma =>
                                              patt_sym
                                                sigma
                                          | phi1 $
                                            phi2 =>
                                              free_svar_subst
                                                psi X
                                                phi1 $
                                              free_svar_subst
                                                psi X
                                                phi2
                                          | ⊥ => ⊥
                                          | phi1 --->
                                            phi2 =>
                                              free_svar_subst
                                                psi X
                                                phi1 --->
                                              free_svar_subst
                                                psi X
                                                phi2
                                          | ex , phi' =>
                                              ex ,
                                              free_svar_subst
                                                psi X
                                                phi'
                                          | mu , phi' =>
                                              mu ,
                                              free_svar_subst
                                                psi X
                                                phi'
                                          end) psi X
                                         phi)} intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember (evar_fresh (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        assert(evar_is_fresh_in (fresh_evar (phi^[[svar: X ↦ psi]])) (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        assert(fresh ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: fresh ∉ B
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
fresh ∉ B
          subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: S (size phi) ≤ sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
      ∪ free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi apply set_evar_fresh_is_fresh'.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: fresh ∉ B
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        subst fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh (elements B) ∉ B
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
  ∪ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} apply not_elem_of_union in H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi)
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} destruct H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        apply not_elem_of_union in H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: (evar_fresh
   (elements
      (free_evars phi^[[svar:X↦psi]]
       ∪ free_evars phi ∪ 
       free_evars psi))
 ∉ free_evars phi^[[svar:X↦psi]])
∧ evar_fresh
    (elements
       (free_evars phi^[[svar:X↦psi]]
        ∪ free_evars phi ∪ 
        free_evars psi)) ∉ 
  free_evars phi
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)} destruct H3.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
H3: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi))
∉ free_evars phi^[[svar:X↦psi]]
H5: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars phi
H4: evar_fresh
  (elements
     (free_evars phi^[[svar:X↦psi]]
      ∪ free_evars phi ∪ 
      free_evars psi)) ∉ 
free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember ((free_evars (phi^[[svar: X ↦ psi]])) ∪ (free_evars phi) ∪ (free_evars psi)) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
H3: evar_fresh (elements B)
∉ free_evars phi^[[svar:X↦psi]]
H5: evar_fresh (elements B) ∉ free_evars phi
H4: evar_fresh (elements B) ∉ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        remember (evar_fresh (elements B)) as fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        assert(evar_is_fresh_in fresh (phi^[[svar: X ↦ psi]])).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) 
              (X : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦
    fresh_evar
      ((fix free_svar_subst
          (psi : Pattern) 
          (X : svar) (phi : Pattern) {struct phi} :
            Pattern :=
          match phi with
          | patt_free_evar x => patt_free_evar x
          | patt_free_svar X' =>
              if decide (X = X')
              then psi
              else patt_free_svar X'
          | patt_bound_evar x => patt_bound_evar x
          | patt_bound_svar X' => patt_bound_svar X'
          | patt_sym sigma => patt_sym sigma
          | phi1 $ phi2 =>
              free_svar_subst psi X phi1 $
              free_svar_subst psi X phi2
          | ⊥ => ⊥
          | phi1 ---> phi2 =>
              free_svar_subst psi X phi1 --->
              free_svar_subst psi X phi2
          | ex , phi' =>
              ex , free_svar_subst psi X phi'
          | mu , phi' =>
              mu , free_svar_subst psi X phi'
          end) psi X phi)}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
fresh ∉ free_evars phi^[[svar:X↦psi]] assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        epose (eval_fresh_evar_open _ _ _ c 0 ρ
                                              H2 H3) as HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
HFresh:= eval_fresh_evar_open phi^[[svar:X↦psi]]
  (fresh_evar phi^[[svar:X↦psi]]) fresh c 0 ρ
  H2 H3: eval
  (update_evar_val
     (fresh_evar phi^[[svar:X↦psi]]) c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦
  fresh_evar phi^[[svar:X↦psi]]} =
eval (update_evar_val fresh c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦fresh}
x
∈ eval
    (update_evar_val
       (fresh_evar
          ((fix free_svar_subst
              (psi : Pattern) (X : svar)
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x => patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X = X')
                  then psi
                  else patt_free_svar X'
              | patt_bound_evar x => patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma => patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi X phi1 $
                  free_svar_subst psi X phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi X phi1 --->
                  free_svar_subst psi X phi2
              | ex , phi' =>
                  ex , free_svar_subst psi X phi'
              | mu , phi' =>
                  mu , free_svar_subst psi X phi'
              end) psi X phi)) c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh_evar
                                  ((fix
                                      free_svar_subst
                                      (psi : Pattern)
                                      (X : svar)
                                      (phi : Pattern)
                                      {struct phi} :
                                        Pattern :=
                                      match phi with
                                      | patt_free_evar
                                        x =>
                                          patt_free_evar
                                            x
                                      | patt_free_svar
                                        X' =>
                                          if
                                           decide
                                             (X = X')
                                          then psi
                                          else
                                           patt_free_svar
                                             X'
                                      | patt_bound_evar
                                        x =>
                                          patt_bound_evar
                                            x
                                      | patt_bound_svar
                                        X' =>
                                          patt_bound_svar
                                            X'
                                      | patt_sym sigma =>
                                          patt_sym
                                            sigma
                                      | phi1 $ phi2 =>
                                          free_svar_subst
                                            psi X phi1 $
                                          free_svar_subst
                                            psi X phi2
                                      | ⊥ => ⊥
                                      | phi1 ---> phi2 =>
                                          free_svar_subst
                                            psi X phi1 --->
                                          free_svar_subst
                                            psi X phi2
                                      | ex , phi' =>
                                          ex ,
                                          free_svar_subst
                                            psi X phi'
                                      | mu , phi' =>
                                          mu ,
                                          free_svar_subst
                                            psi X phi'
                                      end) psi X phi)}
        fold free_svar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
HFresh:= eval_fresh_evar_open phi^[[svar:X↦psi]]
  (fresh_evar phi^[[svar:X↦psi]]) fresh c 0 ρ
  H2 H3: eval
  (update_evar_val
     (fresh_evar phi^[[svar:X↦psi]]) c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦
  fresh_evar phi^[[svar:X↦psi]]} =
eval (update_evar_val fresh c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦fresh}
x
∈ eval
    (update_evar_val (fresh_evar phi^[[svar:X↦psi]]) c
       ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh_evar
                                 phi^[[svar:X↦psi]]}
        rewrite -> HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
HFresh:= eval_fresh_evar_open phi^[[svar:X↦psi]]
  (fresh_evar phi^[[svar:X↦psi]]) fresh c 0 ρ
  H2 H3: eval
  (update_evar_val
     (fresh_evar phi^[[svar:X↦psi]]) c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦
  fresh_evar phi^[[svar:X↦psi]]} =
eval (update_evar_val fresh c ρ)
  phi^[[svar:X↦psi]]^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        clear HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        assert(evar_is_fresh_in fresh phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
evar_is_fresh_in fresh phi
          unfold evar_is_fresh_in.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
fresh ∉ free_evars phi assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        assert(evar_is_fresh_in (fresh_evar phi) phi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
evar_is_fresh_in (fresh_evar phi) phi
          apply set_evar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        epose proof (eval_fresh_evar_open _ _ _ c 0 (update_svar_val X (eval ρ psi) ρ) _ _) as HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
HFresh: eval
  (update_evar_val ?x c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?x} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  ?ϕ^{evar:0↦?y}
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        rewrite -> HFresh in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
HFresh: eval
  (update_evar_val (fresh_evar phi) c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦fresh_evar phi} =
eval
  (update_evar_val ?y c
     (update_svar_val X (eval ρ psi) ρ))
  phi^{evar:0↦?y}
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        clear HFresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        epose (IHsz (phi^{evar: 0 ↦ fresh}) 
                    psi X 
                    (update_evar_val fresh c ρ) _ _ ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal15 ?Goal16: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        pose (eval_free_evar_independent ρ fresh c psi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal15 ?Goal16: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
        rewrite -> e0 in e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal15 ?Goal16: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi clear e0.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal15 ?Goal16: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        rewrite -> (evar_open_free_svar_subst_comm).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    phi^{evar:0↦fresh}^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal15 ?Goal16: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        unfold free_svar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) (X : svar) (phi : Pattern)
        {struct phi} : Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' => ex , free_svar_subst psi X phi'
        | mu , phi' => mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh})
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal15 ?Goal16: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        rewrite -> e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_svar_val X (eval ρ psi)
       (update_evar_val fresh c ρ)) phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
evar_is_fresh_in fresh phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e:= IHsz phi^{evar:0↦fresh} psi X
  (update_evar_val fresh c ρ) 
  ?Goal15 ?Goal16: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X
       (eval (update_evar_val fresh c ρ) psi)
       (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
e0:= eval_free_evar_independent ρ fresh c psi: evar_is_fresh_in fresh psi
→ eval (update_evar_val fresh c ρ) psi =
  eval ρ psi
evar_is_fresh_in fresh psi
        all: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_svar_val X (eval ρ psi)
       (update_evar_val fresh c ρ)) phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val ?y c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦?y}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
x
∈ eval
    (update_svar_val X (eval ρ psi)
       (update_evar_val fresh c ρ)) phi^{evar:0↦fresh} exact H1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh} apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (ex , phi) 0 = true
Hwfc2: well_formed_closed_ex_aux (ex , phi) 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh} simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed phi^{evar:0↦fresh}
          apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true
∧ well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 =
  true split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 = true
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_mu_aux phi^{evar:0↦fresh} 0 = true now apply wfc_mu_aux_body_ex_imp1.
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 0 = true
Hwfc2: well_formed_closed_ex_aux phi 1 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux phi^{evar:0↦fresh} 0 = true now apply wfc_ex_aux_body_ex_imp1.
        *
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply andb_true_iff in Hwf as [_ Hwf].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed psi = true
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply andb_true_iff in Hwf as [_ Hwf].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed_ex_aux psi 0 = true
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
e: well_formed_closed phi^{evar:0↦fresh}
→ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi^{evar:0↦fresh}) =
  eval
    (update_svar_val X 
       (eval ρ psi) (update_evar_val fresh c ρ))
    phi^{evar:0↦fresh}
well_formed_closed_ex_aux psi 0 apply Hwf.
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
eval ρ
  (mu ,
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (mu , phi) repeat rewrite -> eval_mu_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
(let X0 :=
   fresh_svar
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi) in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X0 S ρ in
      eval ρ'
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x => patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x => patt_bound_evar x
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{svar:0↦X0})) =
(let X0 := fresh_svar phi in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' :=
        update_svar_val X0 S
          (update_svar_val X (eval ρ psi) ρ) in
      eval ρ' phi^{svar:0↦X0})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar
             ((fix free_svar_subst
                 (psi : Pattern) (X : svar)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x =>
                     patt_free_evar x
                 | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                 | patt_bound_evar x =>
                     patt_bound_evar x
                 | patt_bound_svar X' =>
                     patt_bound_svar X'
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                 | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                 | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                 end) psi X phi)) S ρ)
       ((fix free_svar_subst
           (psi : Pattern) (X : svar) (phi : Pattern)
           {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x => patt_free_evar x
           | patt_free_svar X' =>
               if decide (X = X')
               then psi
               else patt_free_svar X'
           | patt_bound_evar x => patt_bound_evar x
           | patt_bound_svar X' => patt_bound_svar X'
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               free_svar_subst psi X phi1 $
               free_svar_subst psi X phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               free_svar_subst psi X phi1 --->
               free_svar_subst psi X phi2
           | ex , phi' =>
               ex , free_svar_subst psi X phi'
           | mu , phi' =>
               mu , free_svar_subst psi X phi'
           end) psi X phi)^{svar:0↦fresh_svar
                                     ((fix
                                         free_svar_subst
                                         (psi : Pattern)
                                         (X : svar)
                                         (phi : Pattern)
                                         {struct phi} :
                                           Pattern :=
                                         match
                                           phi
                                         with
                                         | patt_free_evar
                                           x =>
                                             patt_free_evar
                                               x
                                         | patt_free_svar
                                           X' =>
                                             if
                                              decide
                                                (X =
                                                 X')
                                             then psi
                                             else
                                              patt_free_svar
                                                X'
                                         | patt_bound_evar
                                           x =>
                                             patt_bound_evar
                                               x
                                         | patt_bound_svar
                                           X' =>
                                             patt_bound_svar
                                               X'
                                         | patt_sym
                                           sigma =>
                                             patt_sym
                                               sigma
                                         | phi1 $ phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 $
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ⊥ => ⊥
                                         | phi1 --->
                                           phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 --->
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ex , phi' =>
                                             ex ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         | mu , phi' =>
                                             mu ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         end) psi X
                                        phi)}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      fold free_svar_subst in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                    phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      assert ((λ S : propset (Domain M),
                     eval (update_svar_val (fresh_svar (phi^[[svar: X ↦ psi]])) S ρ)
                          ((phi^[[svar: X ↦ psi]])^{svar: 0 ↦ fresh_svar (phi^[[svar: X ↦ psi]])}))
              =
              (λ S : propset (Domain M),
                     eval (update_svar_val (fresh_svar phi) S (update_svar_val X (eval ρ psi) ρ))
                                            (phi^{svar: 0 ↦ fresh_svar phi}))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
(λ S : propset M,
   eval
     (update_svar_val (fresh_svar phi^[[svar:X↦psi]])
        S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                  phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
H: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦
       fresh_svar phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
∀ x : propset M,
  eval
    (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x
       ρ)
    phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                 phi^[[svar:X↦psi]]} =
  eval
    (update_svar_val (fresh_svar phi) x
       (update_svar_val X (eval ρ psi) ρ))
    phi^{svar:0↦fresh_svar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
H: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦
       fresh_svar phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
H: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦
       fresh_svar phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      + (*Create a common fresh var.*)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
        remember ((free_svars phi) ∪ (free_svars psi) ∪ (free_svars (phi^[[svar: X ↦ psi]])) ∪ 
                                   (free_svars (patt_free_svar X))) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
        remember (svar_fresh (elements B)) as MuZ.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
        remember (fresh_svar phi) as MuX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        remember (fresh_svar (phi^[[svar: X ↦ psi]])) as MuY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        assert(MuZ ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
MuZ ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ B
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
MuZ ∉ B
          subst.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ phi0^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
x: propset M
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
∉ free_svars phi ∪ free_svars psi
  ∪ free_svars phi^[[svar:X↦psi]]
  ∪ free_svars (patt_free_svar X) apply set_svar_fresh_is_fresh'.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ B
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ
∉ free_svars phi ∪ free_svars psi
  ∪ free_svars phi^[[svar:X↦psi]]
  ∪ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} apply not_elem_of_union in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: (MuZ
 ∉ free_svars phi ∪ free_svars psi
   ∪ free_svars phi^[[svar:X↦psi]])
∧ MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} destruct H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ
∉ free_svars phi ∪ free_svars psi
  ∪ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        apply not_elem_of_union in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: (MuZ ∉ free_svars phi ∪ free_svars psi)
∧ MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} destruct H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi ∪ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        apply not_elem_of_union in H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: (MuZ ∉ free_svars phi) ∧ MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} destruct H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        erewrite -> (eval_fresh_svar_open _ MuX MuZ); try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuZ x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        erewrite -> (eval_fresh_svar_open _ MuY MuZ); try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuZ} =
eval
  (update_svar_val MuZ x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        erewrite -> svar_open_free_svar_subst_comm; try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val MuZ x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        rewrite update_svar_val_comm; try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi 
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
          simpl in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ {[X]}
MuZ ≠ X apply not_elem_of_singleton_1 in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ≠ X
MuZ ≠ X assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        epose (IHsz (phi^{svar: 0 ↦ MuZ}) psi X 
                    (update_svar_val MuZ x ρ) _ _ _).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal21 ?Goal22 ?Goal23: eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        rewrite e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal21 ?Goal22 ?Goal23: eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ} =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        erewrite (eval_free_svar_independent _ MuZ x psi).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal21 ?Goal22 ?Goal23: eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ} =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal21 ?Goal22 ?Goal23: eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
svar_is_fresh_in MuZ psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal21 ?Goal22 ?Goal23: eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
svar_is_fresh_in MuZ psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        all: auto.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
          apply andb_true_iff in Hwf as [_ Hwf].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed psi = true
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0 apply andb_true_iff in Hwf as [Hwf _].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed_closed_mu_aux psi 0 = true
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0 apply Hwf.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
          simpl in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ {[X]}
MuZ ≠ X apply not_elem_of_singleton_1 in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ≠ X
MuZ ≠ X assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
          subst MuY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in (fresh_svar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]] apply set_svar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
          subst MuX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in (fresh_svar phi) phi apply set_svar_fresh_is_fresh.
        }
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
H: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                    phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) fold free_svar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
H: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                    phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) rewrite H.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
H: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) reflexivity.
        (* replace 1 with (0 + 1) by lia.
        rewrite -free_svar_subst_nest_mu_1.
        rewrite nest_mu_aux_wfc_mu.
        { unfold well_formed,well_formed_closed in Hwf. destruct_and!. assumption. }
        rewrite H. reflexivity. *)
    -
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
eval ρ
  (mu ,
   (fix free_svar_subst
      (psi : Pattern) (X : svar) (phi : Pattern)
      {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x => patt_free_evar x
      | patt_free_svar X' =>
          if decide (X = X')
          then psi
          else patt_free_svar X'
      | patt_bound_evar x => patt_bound_evar x
      | patt_bound_svar X' => patt_bound_svar X'
      | patt_sym sigma => patt_sym sigma
      | phi1 $ phi2 =>
          free_svar_subst psi X phi1 $
          free_svar_subst psi X phi2
      | ⊥ => ⊥
      | phi1 ---> phi2 =>
          free_svar_subst psi X phi1 --->
          free_svar_subst psi X phi2
      | ex , phi' => ex , free_svar_subst psi X phi'
      | mu , phi' => mu , free_svar_subst psi X phi'
      end) psi X phi) =
eval (update_svar_val X (eval ρ psi) ρ) (mu , phi) repeat rewrite -> eval_mu_simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
(let X0 :=
   fresh_svar
     ((fix free_svar_subst
         (psi : Pattern) (X : svar) (phi : Pattern)
         {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x => patt_free_evar x
         | patt_free_svar X' =>
             if decide (X = X')
             then psi
             else patt_free_svar X'
         | patt_bound_evar x => patt_bound_evar x
         | patt_bound_svar X' => patt_bound_svar X'
         | patt_sym sigma => patt_sym sigma
         | phi1 $ phi2 =>
             free_svar_subst psi X phi1 $
             free_svar_subst psi X phi2
         | ⊥ => ⊥
         | phi1 ---> phi2 =>
             free_svar_subst psi X phi1 --->
             free_svar_subst psi X phi2
         | ex , phi' =>
             ex , free_svar_subst psi X phi'
         | mu , phi' =>
             mu , free_svar_subst psi X phi'
         end) psi X phi) in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' := update_svar_val X0 S ρ in
      eval ρ'
        ((fix free_svar_subst
            (psi : Pattern) (X : svar) (phi : Pattern)
            {struct phi} : Pattern :=
            match phi with
            | patt_free_evar x => patt_free_evar x
            | patt_free_svar X' =>
                if decide (X = X')
                then psi
                else patt_free_svar X'
            | patt_bound_evar x => patt_bound_evar x
            | patt_bound_svar X' => patt_bound_svar X'
            | patt_sym sigma => patt_sym sigma
            | phi1 $ phi2 =>
                free_svar_subst psi X phi1 $
                free_svar_subst psi X phi2
            | ⊥ => ⊥
            | phi1 ---> phi2 =>
                free_svar_subst psi X phi1 --->
                free_svar_subst psi X phi2
            | ex , phi' =>
                ex , free_svar_subst psi X phi'
            | mu , phi' =>
                mu , free_svar_subst psi X phi'
            end) psi X phi)^{svar:0↦X0})) =
(let X0 := fresh_svar phi in
 LeastFixpointOf
   (λ S : propset M,
      let ρ' :=
        update_svar_val X0 S
          (update_svar_val X (eval ρ psi) ρ) in
      eval ρ' phi^{svar:0↦X0})) simpl.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar
             ((fix free_svar_subst
                 (psi : Pattern) (X : svar)
                 (phi : Pattern) {struct phi} :
                   Pattern :=
                 match phi with
                 | patt_free_evar x =>
                     patt_free_evar x
                 | patt_free_svar X' =>
                     if decide (X = X')
                     then psi
                     else patt_free_svar X'
                 | patt_bound_evar x =>
                     patt_bound_evar x
                 | patt_bound_svar X' =>
                     patt_bound_svar X'
                 | patt_sym sigma => patt_sym sigma
                 | phi1 $ phi2 =>
                     free_svar_subst psi X phi1 $
                     free_svar_subst psi X phi2
                 | ⊥ => ⊥
                 | phi1 ---> phi2 =>
                     free_svar_subst psi X phi1 --->
                     free_svar_subst psi X phi2
                 | ex , phi' =>
                     ex , free_svar_subst psi X phi'
                 | mu , phi' =>
                     mu , free_svar_subst psi X phi'
                 end) psi X phi)) S ρ)
       ((fix free_svar_subst
           (psi : Pattern) (X : svar) (phi : Pattern)
           {struct phi} : Pattern :=
           match phi with
           | patt_free_evar x => patt_free_evar x
           | patt_free_svar X' =>
               if decide (X = X')
               then psi
               else patt_free_svar X'
           | patt_bound_evar x => patt_bound_evar x
           | patt_bound_svar X' => patt_bound_svar X'
           | patt_sym sigma => patt_sym sigma
           | phi1 $ phi2 =>
               free_svar_subst psi X phi1 $
               free_svar_subst psi X phi2
           | ⊥ => ⊥
           | phi1 ---> phi2 =>
               free_svar_subst psi X phi1 --->
               free_svar_subst psi X phi2
           | ex , phi' =>
               ex , free_svar_subst psi X phi'
           | mu , phi' =>
               mu , free_svar_subst psi X phi'
           end) psi X phi)^{svar:0↦fresh_svar
                                     ((fix
                                         free_svar_subst
                                         (psi : Pattern)
                                         (X : svar)
                                         (phi : Pattern)
                                         {struct phi} :
                                           Pattern :=
                                         match
                                           phi
                                         with
                                         | patt_free_evar
                                           x =>
                                             patt_free_evar
                                               x
                                         | patt_free_svar
                                           X' =>
                                             if
                                              decide
                                                (X =
                                                 X')
                                             then psi
                                             else
                                              patt_free_svar
                                                X'
                                         | patt_bound_evar
                                           x =>
                                             patt_bound_evar
                                               x
                                         | patt_bound_svar
                                           X' =>
                                             patt_bound_svar
                                               X'
                                         | patt_sym
                                           sigma =>
                                             patt_sym
                                               sigma
                                         | phi1 $ phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 $
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ⊥ => ⊥
                                         | phi1 --->
                                           phi2 =>
                                             free_svar_subst
                                               psi X
                                               phi1 --->
                                             free_svar_subst
                                               psi X
                                               phi2
                                         | ex , phi' =>
                                             ex ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         | mu , phi' =>
                                             mu ,
                                             free_svar_subst
                                               psi X
                                               phi'
                                         end) psi X
                                        phi)}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      fold free_svar_subst in *.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                    phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      assert ((λ S : propset (Domain M),
                     eval (update_svar_val (fresh_svar (phi^[[svar: X ↦ psi]])) S ρ)
                          ((phi^[[svar: X ↦ psi]])^{svar: 0 ↦ fresh_svar (phi^[[svar: X ↦ psi]])}))
             =
              (λ S : propset (Domain M),
                     eval (update_svar_val (fresh_svar phi) S (update_svar_val X (eval ρ psi) ρ))
                          (phi^{svar: 0 ↦ fresh_svar phi}))).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
(λ S : propset M,
   eval
     (update_svar_val (fresh_svar phi^[[svar:X↦psi]])
        S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                  phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
H1: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦
       fresh_svar phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      apply functional_extensionality.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
∀ x : propset M,
  eval
    (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x
       ρ)
    phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                 phi^[[svar:X↦psi]]} =
  eval
    (update_svar_val (fresh_svar phi) x
       (update_svar_val X (eval ρ psi) ρ))
    phi^{svar:0↦fresh_svar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
H1: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦
       fresh_svar phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) intros.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
H1: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦
       fresh_svar phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val 
          (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi})
      + (*Create a common fresh var.*)
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
        remember ((free_svars phi) ∪ (free_svars psi) ∪ (free_svars (phi^[[svar: X ↦ psi]])) ∪ 
                                   (free_svars (patt_free_svar X))) as B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
        remember (svar_fresh (elements B)) as MuZ.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val (fresh_svar phi) x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦fresh_svar phi}
        remember (fresh_svar phi) as MuX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
eval
  (update_svar_val (fresh_svar phi^[[svar:X↦psi]]) x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                               phi^[[svar:X↦psi]]} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        remember (fresh_svar (phi^[[svar: X ↦ psi]])) as MuY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        assert(MuZ ∉ B).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
MuZ ∉ B
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ B
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
MuZ ∉ B
          subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: S (size phi) ≤ sz
x: propset M
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
∉ free_svars phi ∪ free_svars psi
  ∪ free_svars phi^[[svar:X↦psi]]
  ∪ free_svars (patt_free_svar X) apply set_svar_fresh_is_fresh'.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
B: SVarSet
HeqB: B =
free_svars phi ∪ free_svars psi
∪ free_svars phi^[[svar:X↦psi]]
∪ free_svars (patt_free_svar X)
MuZ: svar
HeqMuZ: MuZ = svar_fresh (elements B)
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ B
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        subst B.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ
∉ free_svars phi ∪ free_svars psi
  ∪ free_svars phi^[[svar:X↦psi]]
  ∪ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} apply not_elem_of_union in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: (MuZ
 ∉ free_svars phi ∪ free_svars psi
   ∪ free_svars phi^[[svar:X↦psi]])
∧ MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} destruct H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ
∉ free_svars phi ∪ free_svars psi
  ∪ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        apply not_elem_of_union in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: (MuZ ∉ free_svars phi ∪ free_svars psi)
∧ MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} destruct H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi ∪ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}
        apply not_elem_of_union in H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: (MuZ ∉ free_svars phi) ∧ MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX} destruct H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuX x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuX}

        erewrite -> (eval_fresh_svar_open _ MuX MuZ); try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuY x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuY} =
eval
  (update_svar_val MuZ x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        erewrite -> (eval_fresh_svar_open _ MuY MuZ); try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^[[svar:X↦psi]]^{svar:0↦MuZ} =
eval
  (update_svar_val MuZ x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        erewrite -> svar_open_free_svar_subst_comm; try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val MuZ x
     (update_svar_val X (eval ρ psi) ρ))
  phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        rewrite update_svar_val_comm; try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi 
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
          simpl in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ {[X]}
MuZ ≠ X apply not_elem_of_singleton_1 in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ≠ X
MuZ ≠ X assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        epose (IHsz (phi^{svar: 0 ↦ MuZ}) psi X 
                    (update_svar_val MuZ x ρ) _ _ ).
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal23 ?Goal24: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
eval (update_svar_val MuZ x ρ)
  phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        rewrite e.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal23 ?Goal24: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal23 ?Goal24: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ} =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi 
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal23 ?Goal24: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed phi^{svar:0↦MuZ}
          apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (mu , phi) 0 = true
Hwfc2: well_formed_closed_ex_aux (mu , phi) 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed phi^{svar:0↦MuZ} simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed phi^{svar:0↦MuZ}
          apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed_mu_aux phi^{svar:0↦MuZ} 0 = true
∧ well_formed_closed_ex_aux phi^{svar:0↦MuZ} 0 = true split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed_mu_aux phi^{svar:0↦MuZ} 0 = true
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed_ex_aux phi^{svar:0↦MuZ} 0 = true
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed_mu_aux phi^{svar:0↦MuZ} 0 = true now apply wfc_mu_aux_body_mu_imp1.
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
well_formed_closed_ex_aux phi^{svar:0↦MuZ} 0 = true now apply wfc_ex_aux_body_mu_imp1.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal24 ?Goal25: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
eval
  (update_svar_val X
     (eval (update_svar_val MuZ x ρ) psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ} =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        erewrite (eval_free_svar_independent _ MuZ x psi); try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
e:= IHsz phi^{svar:0↦MuZ} psi X
  (update_svar_val MuZ x ρ) 
  ?Goal23 ?Goal24: well_formed_closed phi^{svar:0↦MuZ}
→ eval (update_svar_val MuZ x ρ)
    phi^{svar:0↦MuZ}^[[svar:X↦psi]] =
  eval
    (update_svar_val X
       (eval (update_svar_val MuZ x ρ) psi)
       (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ} =
eval
  (update_svar_val X (eval ρ psi)
     (update_svar_val MuZ x ρ)) phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        reflexivity.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        unfold well_formed,well_formed_closed in Hwf.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: [&& well_formed_positive psi,
    well_formed_closed_mu_aux psi 0
  & well_formed_closed_ex_aux psi 0]
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        destruct_and!.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
H5: well_formed_positive psi = true
H7: well_formed_closed_mu_aux psi 0 = true
H8: well_formed_closed_ex_aux psi 0 = true
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux psi 0
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi all: try assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
MuZ ≠ X
          simpl in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ {[X]}
MuZ ≠ X apply not_elem_of_singleton_1 in H2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ≠ X
MuZ ≠ X assumption.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuY phi^[[svar:X↦psi]]
          subst MuY.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in (fresh_svar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]] apply set_svar_fresh_is_fresh.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in MuX phi
          subst MuX.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
svar_is_fresh_in (fresh_svar phi) phi apply set_svar_fresh_is_fresh.
        }
      +
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
H1: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                    phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) fold free_svar_subst.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
H1: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val
          (fresh_svar phi^[[svar:X↦psi]]) S ρ)
       phi^[[svar:X↦psi]]^{svar:0↦fresh_svar
                                    phi^[[svar:X↦psi]]}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) rewrite H1.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
H1: (λ S : propset M,
   eval
     (update_svar_val
        (fresh_svar phi^[[svar:X↦psi]]) S ρ)
     phi^[[svar:X↦psi]]^{svar:0↦
     fresh_svar phi^[[svar:X↦psi]]}) =
(λ S : propset M,
   eval
     (update_svar_val 
        (fresh_svar phi) S
        (update_svar_val X (eval ρ psi) ρ))
     phi^{svar:0↦fresh_svar phi})
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) =
LeastFixpointOf
  (λ S : propset M,
     eval
       (update_svar_val (fresh_svar phi) S
          (update_svar_val X (eval ρ psi) ρ))
       phi^{svar:0↦fresh_svar phi}) reflexivity.
        Unshelve.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi 
        apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' => patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
          subst sz.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ phi0^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ size phi erewrite <- evar_open_size.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ phi0^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
size phi ≤ size phi reflexivity.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H: x
∈ eval (update_evar_val fresh c ρ)
    phi^[[svar:X↦psi]]^{evar:0↦fresh}
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
          subst sz.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi0) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ size phi erewrite <- evar_open_size.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi : Pattern) {struct phi} :
                Pattern :=
              match phi with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi0) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi ≤ size phi reflexivity.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
H0: size phi = sz
c, x: M
H1: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H2: fresh ∉ free_evars phi^[[svar:X↦psi]]
H4: fresh ∉ free_evars phi
H3: fresh ∉ free_evars psi
H5: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H6: evar_is_fresh_in fresh phi
H7: evar_is_fresh_in (fresh_evar phi) phi
H: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi^{evar:0↦fresh} ≤ sz
          erewrite <- evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
size phi ≤ sz lia.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
fresh: evar
H1: x
∈ eval (update_evar_val fresh c ρ)
    ((fix free_svar_subst
        (psi : Pattern) 
        (X : svar) (phi : Pattern) {struct phi} :
          Pattern :=
        match phi with
        | patt_free_evar x => patt_free_evar x
        | patt_free_svar X' =>
            if decide (X = X')
            then psi
            else patt_free_svar X'
        | patt_bound_evar x => patt_bound_evar x
        | patt_bound_svar X' =>
            patt_bound_svar X'
        | patt_sym sigma => patt_sym sigma
        | phi1 $ phi2 =>
            free_svar_subst psi X phi1 $
            free_svar_subst psi X phi2
        | ⊥ => ⊥
        | phi1 ---> phi2 =>
            free_svar_subst psi X phi1 --->
            free_svar_subst psi X phi2
        | ex , phi' =>
            ex , free_svar_subst psi X phi'
        | mu , phi' =>
            mu , free_svar_subst psi X phi'
        end) psi X phi)^{evar:0↦fresh}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in (fresh_evar phi) phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi apply set_evar_fresh_is_fresh.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H1: x
∈ eval
    (update_evar_val (fresh_evar phi) c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh_evar phi}
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
evar_is_fresh_in fresh phi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi^{evar:0↦fresh} ≤ sz
          erewrite <- evar_open_size.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
size phi ≤ sz lia.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ
          ((fix free_svar_subst
              (psi0 : Pattern) 
              (X0 : svar) 
              (phi0 : Pattern) {struct phi0} :
                Pattern :=
              match phi0 with
              | patt_free_evar x =>
                  patt_free_evar x
              | patt_free_svar X' =>
                  if decide (X0 = X')
                  then psi0
                  else patt_free_svar X'
              | patt_bound_evar x =>
                  patt_bound_evar x
              | patt_bound_svar X' =>
                  patt_bound_svar X'
              | patt_sym sigma =>
                  patt_sym sigma
              | phi1 $ phi2 =>
                  free_svar_subst psi0 X0 phi1 $
                  free_svar_subst psi0 X0 phi2
              | ⊥ => ⊥
              | phi1 ---> phi2 =>
                  free_svar_subst psi0 X0 phi1 --->
                  free_svar_subst psi0 X0 phi2
              | ex , phi' =>
                  ex ,
                  free_svar_subst psi0 X0 phi'
              | mu , phi' =>
                  mu ,
                  free_svar_subst psi0 X0 phi'
              end) psi X phi) =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (ex , phi)
m: nat
H0: S (size phi) ≤ sz
H: m = sz
c, x: M
H2: evar_is_fresh_in (fresh_evar phi^[[svar:X↦psi]])
  phi^[[svar:X↦psi]]
B: EVarSet
HeqB: B =
free_evars phi^[[svar:X↦psi]] ∪ free_evars phi
∪ free_evars psi
fresh: evar
Heqfresh: fresh = evar_fresh (elements B)
H3: fresh ∉ free_evars phi^[[svar:X↦psi]]
H5: fresh ∉ free_evars phi
H4: fresh ∉ free_evars psi
H6: evar_is_fresh_in fresh phi^[[svar:X↦psi]]
H7: evar_is_fresh_in fresh phi
H8: evar_is_fresh_in (fresh_evar phi) phi
H1: x
∈ eval
    (update_evar_val fresh c
       (update_svar_val X (eval ρ psi) ρ))
    phi^{evar:0↦fresh}
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
          subst sz.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ phi0^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ size phi erewrite <- svar_open_size.
signature: Signature
M: Model
phi: Pattern
IHsz: ∀ (phi0 psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi0 ≤ size phi
  → well_formed psi
    → well_formed_closed phi0
      → eval ρ phi0^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi0
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S (size phi)
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
size phi ≤ size phi reflexivity.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        assumption.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        {
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc: well_formed_closed (mu , phi)
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ} 
          apply andb_true_iff in Hwfc as [Hwfc1 Hwfc2].
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux (mu , phi) 0 = true
Hwfc2: well_formed_closed_ex_aux (mu , phi) 0 = true
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ} simpl in Hwfc1, Hwfc2.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed phi^{svar:0↦MuZ}
          apply andb_true_iff.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux phi^{svar:0↦MuZ} 0 = true
∧ well_formed_closed_ex_aux phi^{svar:0↦MuZ} 0 = true split.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux phi^{svar:0↦MuZ} 0 = true
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_ex_aux phi^{svar:0↦MuZ} 0 = true
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_mu_aux phi^{svar:0↦MuZ} 0 = true now apply wfc_mu_aux_body_mu_imp1.
          --
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
Hwfc1: well_formed_closed_mu_aux phi 1 = true
Hwfc2: well_formed_closed_ex_aux phi 0 = true
H0: size phi = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H: MuZ ∉ free_svars phi
H3: MuZ ∉ free_svars psi
H2: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars (patt_free_svar X)
well_formed_closed_ex_aux phi^{svar:0↦MuZ} 0 = true now apply wfc_ex_aux_body_mu_imp1.
        }
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ sz
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        subst sz.
signature: Signature
M: Model
m: nat
phi: Pattern
H0: S (size phi) ≤ m
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ m
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S m
Hwf: well_formed psi
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi^{svar:0↦MuZ} ≤ m
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi erewrite <- svar_open_size.
signature: Signature
M: Model
m: nat
phi: Pattern
H0: S (size phi) ≤ m
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ m
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S m
Hwf: well_formed psi
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
size phi ≤ m
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi lia.
signature: Signature
M: Model
sz: nat
IHsz: ∀ (phi psi : Pattern) 
  (X : svar) (ρ : Valuation),
  size phi ≤ sz
  → well_formed psi
    → well_formed_closed phi
      → eval ρ phi^[[svar:X↦psi]] =
        eval (update_svar_val X (eval ρ psi) ρ)
          phi
phi, psi: Pattern
X: svar
ρ: Valuation
Hsz: S (size phi) ≤ S sz
Hwf: well_formed psi
m: nat
H0: S (size phi) ≤ sz
H: m = sz
x: propset M
MuZ: svar
HeqMuZ: MuZ =
svar_fresh
  (elements
     (free_svars phi ∪ free_svars psi
      ∪ free_svars phi^[[svar:X↦psi]]
      ∪ free_svars (patt_free_svar X)))
MuX: svar
HeqMuX: MuX = fresh_svar phi
MuY: svar
HeqMuY: MuY = fresh_svar phi^[[svar:X↦psi]]
H1: MuZ ∉ free_svars phi
H4: MuZ ∉ free_svars psi
H3: MuZ ∉ free_svars phi^[[svar:X↦psi]]
H2: MuZ ∉ free_svars (patt_free_svar X)
well_formed psi
        assumption.
  Qed.

  Lemma free_svar_subst_update_exchange: ∀ phi psi X ρ,
      well_formed psi → well_formed_closed phi → 
      eval ρ (phi^[[svar: X ↦ psi]]) =
      eval (@update_svar_val M X (eval ρ psi) ρ) phi.
signature: Signature
M: Model
∀ (phi psi : Pattern) (X : svar) (ρ : Valuation),
  well_formed psi
  → well_formed_closed phi
    → eval ρ phi^[[svar:X↦psi]] =
      eval (update_svar_val X (eval ρ psi) ρ) phi
  Proof.
signature: Signature
M: Model
∀ (phi psi : Pattern) (X : svar) (ρ : Valuation),
  well_formed psi
  → well_formed_closed phi
    → eval ρ phi^[[svar:X↦psi]] =
      eval (update_svar_val X (eval ρ psi) ρ) phi 
    intros.
signature: Signature
M: Model
phi, psi: Pattern
X: svar
ρ: Valuation
H: well_formed psi
H0: well_formed_closed phi
eval ρ phi^[[svar:X↦psi]] =
eval (update_svar_val X (eval ρ psi) ρ) phi apply Private_free_svar_subst_update_exchange with (sz := size phi).
signature: Signature
M: Model
phi, psi: Pattern
X: svar
ρ: Valuation
H: well_formed psi
H0: well_formed_closed phi
size phi ≤ size phi
signature: Signature
M: Model
phi, psi: Pattern
X: svar
ρ: Valuation
H: well_formed psi
H0: well_formed_closed phi
well_formed psi
signature: Signature
M: Model
phi, psi: Pattern
X: svar
ρ: Valuation
H: well_formed psi
H0: well_formed_closed phi
well_formed_closed phi 
    lia.
signature: Signature
M: Model
phi, psi: Pattern
X: svar
ρ: Valuation
H: well_formed psi
H0: well_formed_closed phi
well_formed psi
signature: Signature
M: Model
phi, psi: Pattern
X: svar
ρ: Valuation
H: well_formed psi
H0: well_formed_closed phi
well_formed_closed phi assumption.
signature: Signature
M: Model
phi, psi: Pattern
X: svar
ρ: Valuation
H: well_formed psi
H0: well_formed_closed phi
well_formed_closed phi assumption.
  Qed.

  (* rho(psi) = empty then C[rho(psi)] = empty *)
  Lemma propagate_context_empty psi ρ C :
    @eval M ρ psi = ∅ ->
    eval ρ (subst_ctx C psi) = ∅.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
eval ρ psi = ∅ → eval ρ (subst_ctx C psi) = ∅
  Proof.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
eval ρ psi = ∅ → eval ρ (subst_ctx C psi) = ∅
    intro Hpsi.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
Hpsi: eval ρ psi = ∅
eval ρ (subst_ctx C psi) = ∅ induction C.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
Hpsi: eval ρ psi = ∅
eval ρ (subst_ctx box psi) = ∅
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
p: Pattern
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
eval ρ (subst_ctx (ctx_app_l C p Prf) psi) = ∅
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
p: Pattern
C: Application_context
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
eval ρ (subst_ctx (ctx_app_r p C Prf) psi) = ∅
    *
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
Hpsi: eval ρ psi = ∅
eval ρ (subst_ctx box psi) = ∅ auto.
    *
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
p: Pattern
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
eval ρ (subst_ctx (ctx_app_l C p Prf) psi) = ∅ simpl.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
p: Pattern
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
eval ρ (subst_ctx C psi $ p) = ∅ rewrite eval_app_simpl.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
p: Pattern
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
app_ext (eval ρ (subst_ctx C psi)) (eval ρ p) = ∅ rewrite IHC.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
C: Application_context
p: Pattern
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
app_ext ∅ (eval ρ p) = ∅ apply app_ext_bot_l.
    *
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
p: Pattern
C: Application_context
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
eval ρ (subst_ctx (ctx_app_r p C Prf) psi) = ∅ simpl.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
p: Pattern
C: Application_context
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
eval ρ (p $ subst_ctx C psi) = ∅ rewrite eval_app_simpl.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
p: Pattern
C: Application_context
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
app_ext (eval ρ p) (eval ρ (subst_ctx C psi)) = ∅ rewrite IHC.
signature: Signature
M: Model
psi: Pattern
ρ: Valuation
p: Pattern
C: Application_context
Prf: well_formed p
Hpsi: eval ρ psi = ∅
IHC: eval ρ (subst_ctx C psi) = ∅
app_ext (eval ρ p) ∅ = ∅ apply app_ext_bot_r.
  Qed.

  (* application to a singleton *)
  Definition rel_of ρ ϕ: Domain M -> propset (Domain M) :=
    λ m₁,
    (app_ext (@eval M ρ ϕ) {[ m₁ ]}).

  Definition is_total_function f (d c : propset (Domain M)) ρ :=
    ∀ (m₁ : Domain M),
      m₁ ∈ d ->
      ∃ (m₂ : Domain M),
        m₂ ∈ c /\
        app_ext (@eval M ρ f) {[ m₁ ]}
        = {[ m₂ ]}.

  Definition total_function_is_injective f (d : propset (Domain M)) ρ :=
    ∀ (m₁ : Domain M),
      m₁ ∈ d ->
      ∀ (m₂ : Domain M),
        m₂ ∈ d ->
        (rel_of ρ f) m₁ = (rel_of ρ f) m₂ ->
        m₁ = m₂.

  Definition is_functional_pattern ϕ ρ :=
    ∃ (m : Domain M), @eval M ρ ϕ = {[ m ]}.

  Lemma functional_pattern_inj ϕ ρ m₁ m₂ :
    is_functional_pattern ϕ ρ ->
    m₁ ∈ @eval M ρ ϕ ->
    m₂ ∈ @eval M ρ ϕ ->
    m₁ = m₂.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂: M
is_functional_pattern ϕ ρ
→ m₁ ∈ eval ρ ϕ → m₂ ∈ eval ρ ϕ → m₁ = m₂
  Proof.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂: M
is_functional_pattern ϕ ρ
→ m₁ ∈ eval ρ ϕ → m₂ ∈ eval ρ ϕ → m₁ = m₂
    intros [m Hm] Hm₁ Hm₂.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂, m: M
Hm: eval ρ ϕ = {[m]}
Hm₁: m₁ ∈ eval ρ ϕ
Hm₂: m₂ ∈ eval ρ ϕ
m₁ = m₂
    rewrite Hm in Hm₁.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂, m: M
Hm: eval ρ ϕ = {[m]}
Hm₂: m₂ ∈ eval ρ ϕ
Hm₁: m₁ ∈ {[m]}
m₁ = m₂ inversion Hm₁.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂, m: M
Hm: eval ρ ϕ = {[m]}
Hm₂: m₂ ∈ eval ρ ϕ
Hm₁: m₁ ∈ {[m]}
H: m = m₁
m₁ = m₂ subst.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂: M
Hm: eval ρ ϕ = {[m₁]}
Hm₂: m₂ ∈ eval ρ ϕ
Hm₁: m₁ ∈ {[m₁]}
m₁ = m₂ clear Hm₁.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂: M
Hm: eval ρ ϕ = {[m₁]}
Hm₂: m₂ ∈ eval ρ ϕ
m₁ = m₂
    rewrite Hm in Hm₂.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂: M
Hm: eval ρ ϕ = {[m₁]}
Hm₂: m₂ ∈ {[m₁]}
m₁ = m₂ inversion Hm₂.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₁, m₂: M
Hm: eval ρ ϕ = {[m₁]}
Hm₂: m₂ ∈ {[m₁]}
H: m₁ = m₂
m₂ = m₂ subst.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₂: M
Hm₂: m₂ ∈ {[m₂]}
Hm: eval ρ ϕ = {[m₂]}
m₂ = m₂ clear Hm₂.
signature: Signature
M: Model
ϕ: Pattern
ρ: Valuation
m₂: M
Hm: eval ρ ϕ = {[m₂]}
m₂ = m₂
    reflexivity.
  Qed.

  End with_model.

End semantics.


Module Notations.
  Notation "M ⊨ᴹ phi" := (satisfies_model M phi) (left associativity, at level 50) : ml_scope.
  (* FIXME this should not be called `satisfies` *)
  Notation "G ⊨ phi" := (satisfies G phi) (left associativity, at level 50) : ml_scope.
  Notation "M ⊨ᵀ Gamma" := (satisfies_theory M Gamma)
                             (left associativity, at level 50) : ml_scope.
End Notations.

(*Module Hints.*)
#[export]
 Hint Resolve M_predicate_impl : core.
#[export]
 Hint Resolve M_predicate_bott : core.
#[export]
 Hint Resolve M_predicate_exists : core.
#[export]
 Hint Extern 4 (M_predicate _ (evar_open _ _ _)) => mlSimpl : core.
#[export]
 Hint Extern 4 (T_predicate _ (evar_open _ _ _)) => mlSimpl : core.
#[export]
 Hint Extern 4 (M_predicate _ (svar_open _ _ _)) => mlSimpl : core.
#[export]
 Hint Extern 4 (T_predicate _ (svar_open _ _ _)) => mlSimpl : core.
#[export]
 Hint Resolve T_predicate_impl_M_predicate : core.
#[export]
 Hint Resolve T_predicate_impl : core.
#[export]
 Hint Resolve T_predicate_bot : core.
(*End Hints.*)

Global Instance app_ext_proper {Σ : Signature} (M : Model)
: Proper ((≡) ==> (≡) ==> (≡)) (@app_ext Σ M).
Σ: Signature
M: Model
Proper (equiv ==> equiv ==> equiv) app_ext
Proof.
Σ: Signature
M: Model
Proper (equiv ==> equiv ==> equiv) app_ext
  intros X X' HXX' Y Y' HYY'.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
app_ext X Y ≡ app_ext X' Y'
  unfold app_ext.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
{[ e | ∃ le re : M,
         le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
≡ {[ e | ∃ le re : M,
           le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
  rewrite set_equiv_subseteq.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
{[ e | ∃ le re : M,
         le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
⊆ {[ e | ∃ le re : M,
           le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
⊆ {[ e | ∃ le re : M,
           le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
  split.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
{[ e | ∃ le re : M,
         le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
⊆ {[ e | ∃ le re : M,
           le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
{[ e | ∃ le re : M,
         le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
⊆ {[ e | ∃ le re : M,
           le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
  {
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
{[ e | ∃ le re : M,
         le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
⊆ {[ e | ∃ le re : M,
           le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
    rewrite elem_of_subseteq.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
∀ x : M,
  x
  ∈ {[ e | ∃ le re : M,
             le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
  → x
    ∈ {[ e | ∃ le re : M,
               le ∈ X'
               ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
    intros x Hx.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x: M
Hx: x
∈ {[ e | ∃ le re : M,
           le ∈ X
           ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
x
∈ {[ e | ∃ le re : M,
           le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
    rewrite elem_of_PropSet in Hx.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x: M
Hx: ∃ le re : M,
  le ∈ X ∧ re ∈ Y ∧ x ∈ app_interp M le re
x
∈ {[ e | ∃ le re : M,
           le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
    rewrite elem_of_PropSet.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x: M
Hx: ∃ le re : M,
  le ∈ X ∧ re ∈ Y ∧ x ∈ app_interp M le re
∃ le re : M,
  le ∈ X' ∧ re ∈ Y' ∧ x ∈ app_interp M le re
    destruct Hx as [le [re [Hle [Hre Hx] ] ] ].
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
∃ le re : M,
  le ∈ X' ∧ re ∈ Y' ∧ x ∈ app_interp M le re
    exists le.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
∃ re : M, le ∈ X' ∧ re ∈ Y' ∧ x ∈ app_interp M le re exists re.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
le ∈ X' ∧ re ∈ Y' ∧ x ∈ app_interp M le re split.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
le ∈ X'
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
re ∈ Y' ∧ x ∈ app_interp M le re
    {
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
le ∈ X' rewrite -HXX'.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
le ∈ X assumption. }
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
re ∈ Y' ∧ x ∈ app_interp M le re
    split.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
re ∈ Y'
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
x ∈ app_interp M le re
    {
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
re ∈ Y' rewrite -HYY'.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
re ∈ Y assumption. }
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X
Hre: re ∈ Y
Hx: x ∈ app_interp M le re
x ∈ app_interp M le re
    assumption.
  }
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
{[ e | ∃ le re : M,
         le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
⊆ {[ e | ∃ le re : M,
           le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
  {
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
{[ e | ∃ le re : M,
         le ∈ X' ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
⊆ {[ e | ∃ le re : M,
           le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
    rewrite elem_of_subseteq.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
∀ x : M,
  x
  ∈ {[ e | ∃ le re : M,
             le ∈ X'
             ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
  → x
    ∈ {[ e | ∃ le re : M,
               le ∈ X
               ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
    intros x Hx.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x: M
Hx: x
∈ {[ e | ∃ le re : M,
           le ∈ X'
           ∧ re ∈ Y' ∧ e ∈ app_interp M le re ]}
x
∈ {[ e | ∃ le re : M,
           le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
    rewrite elem_of_PropSet in Hx.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x: M
Hx: ∃ le re : M,
  le ∈ X' ∧ re ∈ Y' ∧ x ∈ app_interp M le re
x
∈ {[ e | ∃ le re : M,
           le ∈ X ∧ re ∈ Y ∧ e ∈ app_interp M le re ]}
    rewrite elem_of_PropSet.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x: M
Hx: ∃ le re : M,
  le ∈ X' ∧ re ∈ Y' ∧ x ∈ app_interp M le re
∃ le re : M, le ∈ X ∧ re ∈ Y ∧ x ∈ app_interp M le re
    destruct Hx as [le [re [Hle [Hre Hx] ] ] ].
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
∃ le re : M, le ∈ X ∧ re ∈ Y ∧ x ∈ app_interp M le re
    exists le.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
∃ re : M, le ∈ X ∧ re ∈ Y ∧ x ∈ app_interp M le re exists re.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
le ∈ X ∧ re ∈ Y ∧ x ∈ app_interp M le re split.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
le ∈ X
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
re ∈ Y ∧ x ∈ app_interp M le re
    {
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
le ∈ X rewrite HXX'.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
le ∈ X' assumption. }
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
re ∈ Y ∧ x ∈ app_interp M le re
    split.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
re ∈ Y
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
x ∈ app_interp M le re
    {
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
re ∈ Y rewrite HYY'.
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
re ∈ Y' assumption. }
Σ: Signature
M: Model
X, X': propset M
HXX': X ≡ X'
Y, Y': propset M
HYY': Y ≡ Y'
x, le, re: M
Hle: le ∈ X'
Hre: re ∈ Y'
Hx: x ∈ app_interp M le re
x ∈ app_interp M le re
    assumption.
  }
Qed.